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DRAKE, Reuben Carbert, 1934- A DEVELOPMENTAL STUDY OF IDEAL THEORY, The American University, Ph.D., 1976 Mathematics

Xerox University Microfilms,Ann Arbor, Michigan 4S106

0 1976

REUBEN CARBERT DRAKE

ALL RIGHTS RESERVED A DEVELOPMENTAL STUDY OF IDEAL THEORY

by

Reuben Carbert Drake

Submitted to the

Faculty of the College of Arts and Sciences

of The American University

in Partial Fulfillment of

the Requirements for the Degree

of

Doctor of Philosophy

in

Mathematics

Signature of Committee:

Chairman: Dean of the College ^

I S A y - V Datee: w

1976

The American University Washington, D.C. 20016

t h e AMERICAN UNIVERSITY LIBRARY

s - i C > H TO

Adept

1 1 1

Magnanimity = Mary Rarity

1 1 1

Youthfulncss PREFACE

Though there have been many attempts to study the origin and

development of ideals, no one to date has either written a detailed

account of this algebraic structure from its inception to modern times

as an abstract entity in mathematics nor has any one provided a fairly

"complete" account of the mathematics associated with ideals in its

stages of development. We place the word complete in quotation marks

because a thorough account of such mathematics is virtually impossible.

The papers and books written on the mathematics surrounding the

development of ideals is voluminous! It is the intention of the writer

of this dissertation to provide a detailed account of the major

developments in ideal theory from its inception through the decade of

the 1960's and to provide a more complete account of the associated

mathematics.

Ideals grew out of one of the most fertile periods in the history

of mathematics in attempts to solve the tenowned Fermat ' s' Last Theorem which remains unsolved today. This fertile period, which includes most

of the eighteenth and nineteenth centuries, witnessed the writings and

teachings of Euler, M. Noether, Legendre, Abel, Eisenstein, Cauchy,

Lamé, Rummer, Kronecker, Dedekind, FrObenius, Furtwëngler, Hilbert,

Dickson, and Germain, who were all involved with the Fermat problem in

some way. In view of the involvement of these great mathematicians alone, among many others, one can see the Implications as to how so much mathematics could possibly surround ideal theory.

ill The other periods in the development of ideal theory include the

writings of mathematicians who are equally renowned in their areas of

research. In fact, it is only necessary that we mention the names of

these mathematicians and the particular period with which they are

associated; their works are that well known. The fact that so many great

names in mathematics are associated with ideal theory attests to the

fact that ideals show-up as the fundamental "building blocks" in many

areas of mathematics. In the early part of the twentieth century,

Wedderburn, Lasker, E. Noether, Grell, Krull, Gelfond, Kolmogoroff,

Stone, Silov, Zariski, Artin, and van der Waerden became associated with

the development of ideals. Other great mathematicians who have a

connection with ideals in more recent times are Cohen, Seidenberg, Fuchs,

Gilmer, Kruse, Baer, Nesbitt, Thrall, Levitzki, Jacobson, McCoy,

Murdoch, Curtis, Hewitt, Gillman, Henriksen, Kohls, Helraer, Schilling,

Kelly, Sulinski, Gray, Lambek, Chomsky, SchUtzenberger, and Vandiver.

Ideal theory literally cuts across many of the areas of

mathematics. I was very fortunate to have the opportunity of working with Dr. Mary Gray, Dr. Richard Holzsagerand Dr. Judith Sunley who

represent the areas of ring theory and category theory, algebraic

topology and general topology, and analytic number theory and algebraic

number theory respectfully. Without the combined effort of this

particular combination of mathematicians in terms of their most help­

ful advice, their perception, their patience, and their knowledge of

mathematics, this disseration would have been impossible. For this

reason, I am eternally grateful to them. I should like to thank

Ms. Phyllis Levine who undertook the arduous task of typing this long

iv paper. I should like to thank Ms. Patrinka Osten who made most of the final corrections. I should like to thank Ms. Barbara Loheyde and Mr. Rudolph McShane for their assistance with some of the German translations. I should also like to thank Sharon Hauge, Audrey

Crenshaw and my family for giving me moral support throughout the entire period that I was writing this paper. TABLE OF CONTENTS

PAGE

PREFACE...... ili

CHAPTER

I. INTRODUCTION ...... 1

II. IDEALS AND THEIR FOUNDATIONS IN ALGEBRAIC NUMBER THEORY. . 19

1. Fermat's Last Theorem...... 19 2. Kummer and His Ideal Numbers ...... 28 3. Some Fundamental Concepts of Rummer's Ideal Numbers. . 35 4. Application of Ideal Numbers to Fermat's Problem . . . 54 5. A Later Development Concerning Ideal Numbers ...... 55 6. Dedekind and His Ideal Theory...... 59 7. Some Basic Operations on Ideals...... 64 8. Some Applications of Computer Techniques to Fermat's Last Theorem...... 74 9. Wedderburn and His Hypercomplex Number Systems .... 89

III. IDEALS AND THEIR FOUNDATIONS IN ALGEBRAIC GEOMETRY .... 109

1. Kronecker and His Theory of Forms...... 109 2. Some Fundamental Concepts of F o r m s ...... 112 3. Hilbert and His Basis Theorem...... 128 4. Hilbert's Nullstellensatz...... 142 5. Lasker and His Primary Ideals...... 147 6. Some Fundamental Concepts of Algebraic Geometry. . . . 154 7. Macaulay and His Modular Systems...... 159 8. Van der Waerden and His Reduction of Multi-dimensional Ideals...... 169

IV. IDEAL THEORY DURING THE DECADE OF THE 1920'S ...... 176

1. Emmy Noether and Her Decomposition Theorems...... 176 2. Krull and His Application of Ideal Quotients to a Fundamental Theorem of General Ideal Theory..... 190 3. Emmy Noether and Her Isomorphism Theorems...... 196 4. Grell and His Contractions and Extensions of Ideals. . 202 5. Van der Waerden and His Compenents of An Ideal Determined by a Multipllcatively Closed Set..... 216 6. Krull and the Intersection Theorem...... 222 7. Artin and the Minimal Condition...... 224

Vi CHAPTER PAGE

V. IDEALS IN MODERN RING THEORY...... 233

1. Ideals in Commutative Rings ...... 234

Some Characteristics of Simple Points of An Algebraic Variety...... 234 Prime Ideals and Integral Dependence...... 245 Primal Ide a l s ...... 251 Some Results Based on a Classical Theorem of Noether in Ideal Theory...... 254 Some Fundamentals of Rings in Which All Subrings Are Ideals...... 258

2. Ideals in Noncommutative Rings...... 266

Some Aspects of Rings with Minimal Condition on Left Ideals...... 266 Some Fundamentals of Radical I d e a l s ...... 271 On the Radical of a General Ring...... 277 Some Basic Properties of the Jacobson Radical ...... 286 Prime Ideals in Noncommutative Rings...... 298

VI. SOME APPLICATIONS OF IDEAL THEORY...... 311

1. Some Applications of Ideals in Analysis and Topology. . 311

Applications of Boolean Rings to Point-set Topology . . 311 Applications of Ideals to Continuous Functions...... 329 Continuous Functions on Certain Topological Spaces. . 330 Entire Functions on the Field of Complex Numbers. . . 351 Continuous Functions with Banach Space Structure. . . 364

2. Applications of Ideals in Category Theory ...... 368

The Additive Category ...... 368 The Radical of an Additive Category...... 372 The Brown-McCoy Radical of a Category...... 373

APPENDICES...... 378

A. SELECTED TOPICS FROM ALGEBRA...... 378 B. SELECTED TOPICS FROM ALGEBRAIC GEOMETRY ...... 417 C. SELECTED TOPICS FROM ANALYSIS ...... 422 D. SELECTED TOPICS FROM LINEAR ALGE B R A ...... 430 E. SELECTED TOPICS FROM NUMBER THEORY...... 432 F. SELECTED TOPICS FROM TOPOLOGY . 438

BIBLIOGRAPHY...... 447

vii CHAPTER I

INTRODUCTION

The abstract theory of ideals is a branch of modern abstract

algebra which has come into prominence only in fairly recent years in

spite of the fact that this theory has been around since the middle of

the nineteenth century. It was Emmy Noether whose work during the

decade of the 1920*s greatly contributed to this prominence. Ideals,

in many ways, correspond to the normal subgroups of group theory and

are the most convenient and fundamental "building stones" in a large

number of algebraic structures.

It is the general aim of this exposition to investigate the

origin and major developments in the theory of ideals. In so doing,

the author wishes to point out the mathematical problems which led

specifically to establishment of ideals as bona fide algebraic struc­

tures and to point out how the invention of this structure solved the

particular problems. We shall also discuss some of the related

mathematics which developed as a consequence of this theory. Finally,

at the end of the exposition, we should like to cite some of the

applications of ideals in branches of mathematics other than algebra, such as analysis and topology.

Ideal theory, as we shall see in the succeeding chapters,

literally cuts across most of the major branches of mathematics. In

order to establish notation and to maintain common definitions, we have

1 2 included six appendices. They are as follows: Appendix A: Selected

Topics from Algebra, Appendix B: Selected Topics from Algebraic

Geometry, Appendix C; Selected Topics from Analysis, Appendix D;

Selected Topics from Linear Algebra, Appendix E: Selected Topics from

Number Theory, and Appendix F: Selected Topics from Topology. We begin by establishing a working definition of an ideal and citing some examples of this most versatile structure.

DEFINITION I-l. A subring S of a ring R is a subgroup of the additive group of the ring which is, in addition, closed under multiplication.

DEFINITION 1-2. A left ideal a of a ring R is a subring closed under left multiplication by an arbitrary element of R; that is, if reR and aea, then raea. (The Greek alphabet, a, g, y»etc., where possible, will be used exclusively to indicate ideals throughout this paper.)

Remarks

1. A right ideal a of a ring R is a subring of R such that if reR and aea then area.

2. A (two-sided) ideal is a left ideal which is also a right ideal.

3. If R is a commutative ring, then all one-sided (left or right) ideals are (two-sided) ideals.

4. The term ideal will be used to mean two-sided ideal. 3

Examplesi

1. In the ring of all 2x2 real matrices, the set of all matrices of the form

is a right ideal but not a left ideal. (It is interesting to note that this ring has no Ideals.)

2. In the ring of all nxn matrices with entries from the rationalnumbers, for each i = 1, 2, . . ., n, the set of all matrices with zeros except possibly in column i, is a left ideal but not a right ideal. (As in example I, this ring has no ideals.)

3. In the commutative ring of integers Z, the set of all multiples of two,

(2) = {0, ±2, ±4, ±6, +. . .} is anideal. (The set of all multiples of four,

(4) => {0, ±4, ±8, ±. . .} is anideal of the ring (2),)

4. In the commutative ring of all rational numbers Q, the set of all integers Z is a subring of R; however, Z is not an ideal of R.

(This is easily verified by considering a fraction "^cR, that is, a, bsZ and b 0, and an element xeZ. Observe that neither ^e Z nor x ^ eZ.)

5. In the commutative ring R of all real-valued functions, the set

a = {fcR]f(a) = 0, aeR} is an ideal of R , where R is the set of real numbers. 4

The modern theory of ideals has its foundations in two of the major branches of mathematics; algebraic number theory and algebraic geometry. The ideal theory which evolved from algebraic number theory, called multiplicative ideal theory, is an indirect consequence of

Fermat's Last Theorem.

In 1637 Pierre Fermat[50] proposed his now famous last theorem:

THEOREM I-l. X'^+Y" = Z" has no solution in integers X, Y, Z, none zero, if n is an integer greater than 2.

This theorem was studied by the greatest mathematicians of Europe such as Euler, Legendre, Gauss, Abel, Cauchy, Lame, Kummer, Frobenius,

Furtwangler, Dickson and Germain. Gauss and Legendre, at this time, were already involved with their studies of cubic and quadratic reciprocities.

In view of the previous and present research on cubic and quadratic reciprocity such rings as Z(i), Z(w) and Z(/^) were already known to the mathematical community. Lame[126] in 1847 proposed a proof of Fermat's Theorem for certain prime exponents p. This proof involved an assumption of unique factorization in the ring Z(/^).

Kummer[118] was the first to observe the mistake of this assumption since

9 = 3*3 = (2 + /?)(2 - /^).

Ernst Kummer[121, 123] devised a method of recapturing unique factorization in the set of complex integers by the introduction of

"ideal numbers." Chapter II of this exposition is devoted to a discussion of Kummer's theory and its consequences. In addition to 5

■the results of Kummer, who did prove Fermat's theorem for certain

primes. Chapter II contains the work of Dedekind.

Richard Dedekind, a student of Kummer, is the founder of multi­

plicative ideal theory. He is given credit for formulating the

definition of "ideal" as we know it today. Dedekind[34, 36] worked

primarily with the ring of integers in the field of rational numbers

because of the fact that in the field of rational numbers Q if a^^, a^,

. . .,a^ are integers without a non-trivial common divisor, that is,

(a^, a^, . . . a^) = (1);

then there exist Integers 8.^ such that

31^1+328.2+ . ■ '+^n^n “ 1*

Today, fields which satisfy this property are called Dedekind fields.

Though Dedekind did not directly involve himself with Fermat's Last

Theorem, he did establish some very Important properties of his ideals,

which he called divisors. He observed that their properties often

parallel some of the properties of the factorization of integers.

Suppose we consider the integer 60 in the ring ofintegers Z. We

know that

60 = 2^-3*5 = 2‘2*3*5.

Dedekind[33] showed that principal ideals have a similar decomposition.

For example, suppose we consider the principal ideal (60) in Z.

(60) = (2) (2) (3) (5).

Section 7 of Chapter IX is a summary of these basic operations on

Dedekind's ideals.

Fermat's Last Theorem is still an unsolved problem in the mathe­

matical world today. In modern times (during the twentieth century) 6 mathematicians have continued to try to prove this theorem. The advent of the high-speed digital computer gave these modern mathematicians new hope and courage in their attempts to solve Fermat's Theorem.

H. S. Vandiver, an American mathematician, became an authority on

Fermat's Last Theorem. Section 8 of Chapter II is primarily devoted to the attempts of Vandiver[198, 199, 200, 201] and his various colleagues to solve this problèm. It is interesting to note that Vandiver used

Kummer's test for the regularity of a prime number. Kummer[116] had used this test in his proof of Fermat's Last Theorem for regular primes. Though Vandiver's attempt to solve the problem for a general n was fruitless, he did generate a great deal of new information about the number of units, independent sets of units, and fundamental systems of units which are all associated with certain cyclotomie fields. With the use of a computer, he was also able to prove Fermat's Last Theorem for all regular prime numbers p < 4003.

Ideal theory, that is, the ideal theory developed by Dedekind, was somewhat dormant until the beginning of the twentieth century when

Wedderburn established certain results.

In 1908 Joseph Wedderburn[214] wrote a paper on hypercomplex num­ bers which was a major "turning point" in the development of multipli­ cative ideal theory and modern "abstract algebra" in general. An algebra is a ring with additional properties. In view of the fact that algebras are rings, they too contain ideals. Chapter II (Section 9) is concluded by a discussion of Wedderburn's hypercomplex number systems.

In the decade of the 1920's when ideal theory really came into 7 prominence, Artin[6] used Wedderburn's results to establish some most significant results for rings which satisfy the minimal condition on ideals.

Modern ideal theory, as we mentioned much earlier in our dis­ course, has a double origin. The first, which originated in algebraic number theory as a result of attempts to solve Fermat's Last Theorem, was discussed above. The second origin is in algebraic geometry, as it is called today. This second source of ideal theory is a result of studies of certain polynomial rings associated with algebraic geometry.

In general the polynomial ring C[xj^, X 2 , . . . x^], where C is the field of complex numbers, has been studied by methods which depend on the special nature of this ring. Elimination theory was one of the most important methods associated with these studies. Among the names of the mathematicians associated with this aspect of ideal theory, called additive ideal theory, are Hilbert, Kronecker, Lasker, Macauley and van der Waerden. Chapter III is primarily a discussion of the contributions of these mathematicians.

Leopold Kronecker, like Dedekind, was a student of Kummer. Dur­ ing the early part of the 1880's, Kronecker became interested in some of the algebraic theories of Kummer. In particular, he was desirous of putting the divisors of certain algebraic integers in a simple and natural form in such a manner that the properties of Kummer's ideal numbers could take on a more concrete mathematical mold. By a methodical use of indeterminates, Kronecker[105] arrived at an ideal theory which was equivalent to Dedekind's. Kronecker worked with 8 certain functions in indeterminates ^2» • • • ^ which were called forms.

David Hilbert was greatly influenced by Kronecker, In view of this fact it is no coincidence that he also worked with the polynomial functions, forms. He worked with certain collections of forms which he called "modules." These modules were actually polynomial ideals.

Hilbert[86], utilizing these modules, proved the now famous Basis

Theorem;

THEOREM 1-2 (Hilbert's Basis Theorem). If R is a ring such that every ideal of R is finitely generated, then every ideal of the poly­ nomial ring R[x] is also finitely generated.

Hilbert[84] also proved his celebrated "Nullstellensatz" which is closely related to his Basis Theorem and very much entrenched in the fundamentals of algebraic geometry. The formal statement of this theorem is:

THEOREM 1-3 (Hilbert's Nullstellensatz). If f is a polynomial in

K[x^, X2 > . . . x^] which vanishes at all the common zeros of f-j^, f2 ,

. . . f^ inA^(L); the affine space over the extension field L over the field K, then

f® = hj^f-|^+h2f 2 + • • • +hffr for some natural number g, where the h^'s are polynomials in K[x^^, x^ ,

. . . Xj^].

Emanuel Lasker became involved with one of the standard problems in the theory of polynomial rings. This particular problem involved determining whether or not a polynomial f belongs to a given ideal 9

M = (fjL> fg, . . . f^).

This does not imply that a computational decision method was sought but rather a method which gives an insight into the structure of the particular ideal and expresses the relation between the zeros of the ideal and its elements f^. Lasker[132] was the first to determine such a method. His method depends on the decomposition of ideals into primary components. Thus he is responsible for establishing the main results pertinent to primary decomposition and for defining a primary ideal. The main theorem of his work follows:

THEOREM 1-4 (Lasker's Theorem). Any module N is the least common multiple (L.C.M.) of a finite number of primary ideals.

Lasker, like Hilbert, was influenced by the works of Kronecker.

Host of his discoveries represent an extension of Kronecker's work.

Macaulay was also influenced by the works of Kronecker as well as of Hilbert, Lasker, andM. Noether. Francis Macaulay[142, 144, 145], following the general formats of Lasker and Hilbert, made contribu­ tions to the development of additive ideal theory in two of his major papers. In the first paper he explored the polynomial ideal (module) theory already established by Hilbert and Lasker. The second paper is an extension of the first; however, it does contain some original work of significance on unmixed ideals and inverse systems. Section 7 of

Chapter III explores the essence of these papers. Van der Waerden is the link between Kronecker's additive ideal theory and Dedekind's multiplicative ideal theory which merged during the decade of the

1920's. 10

Bartel van der Waerden, already proficient in the polynomial ring theory of algebraic geometry which Kronecker had perpetuated, went to

Gottingen to study with Emmy Noether. He had already received his Ph.D. from the University of Amsterdam and was a mature mathematician by this time. We conclude Chapter III (Section 8) by discussing van der

Waerden's[209] method of reducing multi-dimensional ideals, which in­ volves polynomial ideal theory. Though van der Waerden was a student of E. Noether, we discuss him before we discuss her because of his association with additive ideal theory. Van der Waerden was also a student of Artin. We shall discuss more of van der Waerden's contribu­ tions in Chapter IV. Section 6 of Chapter III on algebraic geometry also includes a discussion of the work of Max Noether[165], one of the early proponents of algebraic geometry and the father of E. Noether.

Since an equation

f(x^L, X2, . . . = 0 represents an (n-1) dimensional algebraic surface in n-dimensional space with coordinates xj^, . . . x^, each new result uncovered con­ cerning polynomial ideals added some new information to the basic theory of algebraic geometry. During the decade of the 1920's, the two theories fused, which fortified the efforts of researchers and brought this theory into prominence in the mathematical world.

We shall see that Dedekind's definition of an ideal was directed to the rings of integers in algebraic number fields while Kronecker's definition was directed to the ring of integers in function fields.

Though it was later proved that their definitions were equivalent. 11

mathematicians of that era treated the two views as separate, distinct

entities. When the additive and multiplicative theories were dis­

covered to have the same concept of an ideal, mathematicians defined

ideals as has been stated in Definition 1-2, which is not dependent on

a particular domain. The concept of an ideal, as well as ideal theory

in general, became an abstract notion primarily due to E, Noether.

Thus, the efforts of both schools of thought were then combined.

Algebraic geometry today has become a study of commutative ring theory

more.or less. We repeat the fact that this fusion occurred in the

1920's.

Ideal theory during the period 1920-1930 was greatly influenced

by four great mathematicians: Grell, Krull, E. Noether, and van der

Waerden. In fact, one can say that the ideal theory of this period was more or less directed by Noether because Grell, Krull, and van der

Waerden were her students and her axiomatic approach to ideal theory

led to the emergence of the abstract notion of an ideal, bringing about

the fusion of the two theories of ideals. Chapter IV is actually a

discussion of the major developments during this decade.

Emmy Noether[164] was the first mathematician to realize the

general structure of rings in which the maximal condition was

satisfied. For this reason, these rings today are called Noetherian

rings. A dedicated disciple of Dedekind, Noether is responsible for

proving all the main results on the existence and partial uniqueness

of normal decompositions. She also proved that the isolated components

of ideals are uniquely determined by their corresponding prime ideals.

All of this was done in her famous publication of 1921, "Idealtheorie 12 in Ringbereichen.” Krull, her student, wrote a paper based on this paper of Noether[164], which extended some of her results to ideal quotients.

Previously, Noether had proven that each ideal can be represented as the least common multiple of a finite number of primary ideals belonging to different prime ideals. Wolfgang Krull[108] demonstrated how this theorem can be applied by the use of ideal quotients. He also applied her "uniqueness" theorem via ideal quotients and defined some of the fundamental concepts of ideal theory in a new way by using ideal quotients.

In 1927, Noether published another paper which exemplified her creative powers. In this paper, which appeared in Mathematische

Annalen, Noether[163] proved some theorems concerning rings which are also applied to other basic algebraic structures such as groups, modules, vector spaces and algebras. These theorems demonstrate

Noether's uncanny ability to analyze the structure of an algebraic object via an axiomatic approach. Thus she is credited for many significant contributions in abstract structure theory in modern mathematics. The structure theorems of her paper of 1927 are called the "Isomorphism Theorems."

THEOREM 1-5 (First Isomorphism Theorem). Let R and S be rings and let f:R-+-S be a ring homomorphism from R onto S. If a = ker f, then

- - S. a 13

THEOREM 1-6 (Second Isomorphism Theorem). Let a and B be subrings of a ring R with a an ideal of R. Then a OB is an ideal of B, ct+B is a subring of R and

g+B ~ B . g g OB

THEOREM 1-7 (Third Isomorphism Theorem). Let g S B — R and let g and B be Ideals of the ring R. Then B/g is an ideal of R/a and

R _2L. = R . A r g

These theorems demonstrate Noether's creative powers, in particular, her ability to utilize an axiomatic approach. Like Krull, Grell, another student, used her work as a basis for his own research.

In 1927, Heinrich Grell[69] published a paper which was a study of the contraction and extension of ideals. He utilized Noether's isomorphism theorems in his developments. Grell also introduced the concept of a ring of quotients in this paper. This was accomplished by the use of a multiplicative system which we shall see later is a most versatile structure.

Van der Waerden[203], whom we discussed earlier in conjunction with his work on polynomial ideals, in 1928 published a paper which was based somewhat on Noether's paper of 1921[164 ]. He used a multiplica- tively closed set (multiplicative system) to determine the components of an ideal. Noether, in her paper, accomplished this by other means.

Krull also discovered the importance of a multiplicatively closed set.

Krull[111], in his paper of 1928, utilized the concept of a multiplicatively closed set to prove his now famous Intersection Theorem. 14

THEOREM 1-8 (Krull's Intersection Theorem). Let a be an ideal of

a Noetherian ring R; then an element x of R belongs to

00 H a l i=l

if and only if we have that

X = ax

for at least one element aea. Further

CO n i=l

is an isolated component of the zero ideal, (0).

Krull was a most prolific mathematical scholar and appeared to be

primarily interested in the abstract theory of ideals. Though other

students of Noether, such as Holzer, Koethe, Deuring, Fitting, Witt,

Tsen, Shoda, Levitzki, and Hermann, made significant contributions to

algebra and ring theory, we must also include in our discussion Artin, who was not a student of Noether.

Emil Artin was one of the most significant contributors to the

development of ideal theory during the decade of the 1920's who was not

a student of Noether. He became very interested in the work of

Wedderburn[214] on hypercomplex number systems. Artin[6] wrote a paper

about rings which satisfy the minimal condition which paralleled the work of Wedderburn. He also was guided by Noether's[164] work on rings which satisfy the maximal condition. Due to his extensive research on

rings that satisfy the minimal condition, a class of rings has been

named in his honor. 15

DEFINITION 1-3. A ring R is said to be left Artinian if every non-empty set «8 of left ideals of R, partially ordered by set inclusion, has a minimal element ; that is, there is an element aetfi such that if

and gcct, then a = 6.

Remarks

1. In an analogous manner, right Artinian and Artinian rings are defined.

2. It can be shown that any finite-dimensional algebra over a field is Artinian. (One should expect this in view of the fact that

Wedderburn's hypercomplex number systems were actually algebras.)

Van der Waerden, whom we discussed earlier in conjunction with his contributions to additive ideal theory, also belongs to this period

(1920-1930) in terras of his many contributions. Having been trained in the Kronecker school of thought, van der Waerden later became a student of both Noether and Artin. He greatly contributed to the fusion of multiplicative ideal theory and additive via his great classic Modern

Algebra. Van der Waerden*s book[205] was perhaps the mathematical world's first textbook on mod e m algebra. This book contains many of

Noether's and Artin's lectures as well as Kronecker's influence through van der Waerden himself. It is interesting to note that van der

Waerden is the only one of these early pioneers in ideal theory who is alive today.

Chapter V is a discussion of ideal theory in modern times. The results on ideals in the modern theory of ideals naturally divide into two categories: those concerned with ideals associated with 16 commutative rings and those concerned with the Ideals of noncommutative rings. As in the case of the ideal theory during the decade of the

1920's, we do not propose to discuss all of the contributions to ideal theory in modern times. The papers and books on ideal theory produced after 1930 are voluminous. We shall discuss in Chapter V only those papers which are particularly significant to the development of ideal theory in modern times.

After the period 1920-1930, much of the work which has been done on polynomial rings and ideals has become a part of commutative ring theory. In fact, algebraic geometry in its most modern form is a study of commutative ring theory. In view of this fact, one can expect that polynomial rings and ideals will form an integral part of our discus­ sion of ideals in the modern commutative theory. Among the contributors to this aspect of ideal theory we find Zariski[220, 221], Cohen and

Seidenberg[26], Cohen[25], Fuchs[54], Gilmer[64, 65, 66] and Kruse

[114].

A noncommutative ring is a more general ring than a commutative one. The results in commutative ring theory, however, are more de­ tailed than those in noncommutative ring theory. The main focus in noncommutative ring theory is structure theory.

The primary object of any aspect of structure theory is to describe some general objects in terms of some simpler ones. A very convenient method of studying structure theory in noncommutative rings is via radicals. In other words, the noncommutative rings become the general objects and radicals become the simpler objects. Since a 17

radical is an ideal, one can say that noncommutative rings are analyzed by certain ideals of these rings.

In Chapter V, we shall point out several notions of a radical which are defined by several mathematicians. In so doing, we shall discuss the advantages and disadvantages of the various radicals. We

shall also discuss any concepts which show any interdependence of the

ideas and notions of radicals. Artin, Nesbitt and Thrall[7], Baer[10],

Levitzki[139, 140], Jacobson[94], McCoy[152], Murdoch[159] and

Curtis[28] are mathematicians among many who made contributions to this aspect of ideal theory.

Though the natural domain for ideal theory is algebra, it has shown up in several of the other branches of mathematics as early as

the decade of the 1930's when it was in its infancy. Chapter VI explores some of these applications and focuses on Applications of

Ideals in Analysis and Topology and Applications of Ideals in Category

Theory.

The applications of ideals in analysis and topology are all dependent on an early paper by Stone[191]. As a result of this paper, which applies Boolean rings to topology, others proceeded to apply ring and ideal theory to spaces of continuous functions, bounded continuous functions, entire functions and spaces of continuous functions with Banach algebra structure. Finally, since an additive category with one object has a ring structure, one can define an ideal of a category. In addition to this, one can define a radical of a 18 category. In particular, one can also define what the Brown-McCoy radical is in this context. CHAPTER II

IDEALS AND THEIR FOUNDATIONS IN

ALGEBRAIC NUMBER THEORY

1, Fermat's Last Theorem

In 1637, the French mathematician Pierre de Fermat[50] claimed a theorem which eventually led to the discovery of ideals as we know them today. The theorem states that the diophantine equation

- gU with integral n > 2 has no solution in integers x, y, z, where x f 0, y f 0, and z f 0.

From the very inception of this theorem mathematicians have con­ sidered it to be such a great challenge that they have spent many years in pursuit of its proof. Dickson[38], Mordell[158] and Vandiver

[199] all point out the fact that among the great mathematicians of the past who have worked on this problem are Euler, Legendre, Gauss,

Abel, Caucny, Dirichlet, Lame, Rummer, Frobenius, Furtwangler, Dickson, and Germain. After many years of fruitless study, many mathematicians began to entertain the question: Did Fermat himself actually have a proof of the theorem in his possession?

Fermat had become interested in the theory of numbers and in particular he was attracted to the part of this branch of mathematics which dealt with the solution of indeterminate equations namely,

Diophantine Analysis. Diophantine Analysis had been named after the

19 20

Greek mathematician Diophantus, who made several studies of these indeterminate equations during the third century. Fermat had in his possession a copy of Diophantus' works which was edited by Bachet[9l and had "scribbled" in the margin of the book the fact that he had proved the theorem but did not have enough space in the margin to record it. Though most mathematicians of that time did not question the credibility of Fermat's assertion due to his past record as a mathematician. Smith[189] cites the fact that Gauss was of the opinion that if Fermat actually had a proof in mind there was surely a mistake in it.

The attempts to solve the problem, however, were not exactly in vain because each attempt did generate a great deal of mathematics.

After all, this period was one of the most fertile periods in the history of mathematics, especially in the theory of numbers.

Gauss[55] had introduced the ring Z[i] in conjunction with his work on biquadratic reciprocity. In addition to Gauss and the above scholars,

Jacobi and Eisenstein belonged to this period. Jacobi[93] had done extensive research on quadratic reciprocity and his Jacobi sums.

Eisenstein[46] had worked with the ring Z[w] in connection with his research on cubic reciprocity. Finally, some mathematicians were successful in constructing proofs for Fermat's Theorem for certain values of n.

The case where n = 2 appears to be most interesting because it generates the so-called "Pythagorean Triplets." We shall use the explanation given by Grosswald[73] for this special case. Let us 21 consider the Diophantine equation

x^+y2 = 2^.

We observe that the following two (sub) cases occur;

Case I; 2 I x*yz

Case II: 2 | x*yz.

Solution for Case n = 2:

Case I. Suppose x, y, z are all odd, that is, x = 2k^+l, y = 2k2+lj z = 2kg+l, where kj^, k 2 » k^sZ. Now observe that

x2+yZ = (2kj+l)2 + (2k2+l)2 which implies that

x2+y2 = 4k|+4kj^+l+Ak^+4k2+l which implies that

x^+y^ = 2[2kj^Cki+l) + ZkgCkg+l) + H which implies that o 2 X +y is even.

But

7?" = (2kg+l)^ which implies that

z^ = Akg+Akg+l which implies that

z- = 2[2kg(kg+l)] + 1 which implies that 9 z is odd.

Hence x^+y^ = is not solvable whenever 2 \ x*yz. 22

Case II. Note that if x = da,y =dB, where d, a, 6 eZ, then

x2+y2 = (da)2 + (d$)2 implies that

which implies that

But

x^+y2 = ^

Hence

implies that

z2 = d2(y2).

2 2 2 2 (if z is a perfect square, then d and a +B are perfect squares) where

thus

z = dy.

Thus we can conclude that if any two of the variables in x^+y^ = z% have a factor in common, then the third variable must have the same factor in common. Hence we consider

(x,y) = (y,z) = (z,x) = 1 .

But

(x,y) = (y;'z) = (z,x) = 1 implies that at most one of the integers x, y, z is even; the other two

then must be odd. Suppose that x = Zk^+l, y = 2 k2 +l, z = Zk^î that is,

X and y are odd while z is even. Then observe that 23

x2+y2 = 4k^4-4k]^+l+4k^+4k2+l

Implies that

x^+y^ = 4(k|+kj^) + 4Ck|+k2) +2, which implies that

x^+y2 E 2(mod 4).

Moreover,

Z = 2kg implies that

z2 = 4k|, which implies that

x^ = O(mod 4).

This is a contradiction since x^+y^ = z^. Hence either x or y is even. Let x be even. But x even implies that y and z are both odd; y and z both odd implies that z+y and z-y are both even because the sum and difference of two odd numbers are both even. Let z+y = 2m and z-y = 2n for m, nez. Now observe that

x^+y^ = z% implies that

[2 = z%-y2. which implies that

x^ - (z+y)(z-y).

This implies that

2 m*2 nî that is, x^ = 4q^ where q^ = mn. But

x+y = 2 m 24 and

z-y = 2n imply that

= -jCx+y) and

n = -jCz-y) .

Thus

(m,n) = 1 because if (m,n) = k>l, then k|m+n = z, and k|m-n = y, that is k|z and k|y. This is a contradiction because (y,z) = 1. Hence (m,n) = 1 and m'n a perfect square imply that m and n are perfect squares. Let m = ot^ and n = 3^ . But

(m,n) = 1 implies that

(a,3) = 1.

Now z = m+n = a^+3^ , y = m-n = a^-3^ and = 4q^ = 4ot^3^. But

= 4a^3^ implies that

X = 2u3j where (a,3) = 1. Now observe that (a,3) = 1 implies that ot and 3 can­ not both be even because if a and 3 are both even then

(0,3) Ï 2.

But a and 6 cannot both be odd because » and 3 both odd implies that z = is even and y = o^-gZ is even. This is a contradiction because (y,z) = 1. Hence either a is even and 3 is odd or o is odd and

3 is even. (Note that this is equivalent to saying that 25

a*g = 2 (mod 4) because if we let a = 2u and g = 2v+l, where u, veZ; that is^a is even and B is odd, then a * 6 = 2u(2v+l) = 4uv+2u = 2(mod 4),) Hence we con- elude that the equation x^+yZ = has primitive solutions of the form

X = 2 ag

y = nZ-gZ

z = aZ+gZ^

where (a,g) = 1 and a * 3 = 2 (mod 4).

One observes that in the solution for the case where n = 2, one does not require any of the so-called "sophisticated mathematics." The arguments for the most part are arithmetical. This is perhaps one of the most important reasons why mathematicians were so persistent in their efforts to solve this problem. As was mentioned earlier, proofs were found for various values of n. Grosswald[73] and Mordell[158] give proofs of Fermat's Theorem for

x*++y‘t =

x^+yZ =

x^+yS = gS and

Jj7+y7 - J,7

The case where n = 4 can be proved very rigorously by elementary methods also. Since = (zZ)2^ it is sufficient to look at another form of the theorem. The form of the theorem which we desire, how­ ever, follows immediately as a corollary. 26

THEOREM IT-1. There is no solution in positive integers of the

equation

= gZ.

Proof; Suppose that x, y, z are such that x>0, y>0, z>0 and

(x,y) = (y,z) = (z,x) = 1 ,

Since all integers are either odd or even, x is of the form 2m or 2m+l,

where roeZ. Hence xZ is of the form 4mZ or 4mZ+4m+l; that is, %Z is of

the form 4m or 4m+l. Thus a number of the form 4m+2 or 4m+3 cannot be

a square. Considering x^+y^ = z ‘* ^ w e observe that x and y cannot both

be odd because the sum of their fourth powers would be of the form 4 m + 2

and this cannot be a square. Hence either x or y must be even. Let y

be even. Now

x^+y^ = zZ

implies that ,

(xZ)Z + (yZ)Z = gZ ^ ...

By utilizing the results for the case n = 2, we conclude that xZ = aZ-^Z,

yZ = 2ab, z = aZ+bZ, where (a,b) = 1, and a and b are not both odd. If

xZ = aZ-yZ, then a cannot be even because b would be odd, which forces

xZ to be of the form 4m+3, which is impossible. Hence

xZ+bZ = &Z,

where b is even, a is odd and (a,b) = (a,x) = (b,x) = 1. Again utiliz­

ing the solution for the case n = 2 , we conclude that x = pZ-qZ, b = 2 pq, a = pZ+qZ, where p and q are prime to each other and are not both odd. Now

yZ = 2 ab 27

implies that

yZ = 4pq (pZ+qZ).

Now

Cp,q) = 1

implies that

(p, pZ+qZ) = (q, pZ+qZ) = 1 and

Cp, pZ+qZ) = (q, pZ+qZ) = 1

implies that p, q, pZ+qZ are perfect squares. Set p = rZ, q = sZ, pZ+qZ = tZ. Hence the values of x, y, z in terras of r, s , t are given by X = r^-s^, y = 2rst, z = ~ r®+6r**s**+s®. Now

z = r®+6r^s^+s® implies that

z > (r*^+s^) > t*' ; tHat is,

z>t^ implies that

t> yz.

Hence if one solution of the equation x'*+y'* = zZ is known for which none of the unknowns x, y, z is zero, another solution (r, s, t) can be found for which none of the unknowns is zero and such that t<

We continue this process so that an infinite number of positive integers t, t^, t2 > . . . ^t^ can be found such that t]^ < 1^, t2 <

. . . which is clearly absurd. Hence, x*^+y‘* = zZ has no solution in positive integers. (Note that the method used in the proof is called the method of infinite descent.) 28

Corollary: There is no solution in positive integers for the equation

,

In spite of the proofs for these special cases of values for n, no one has proved Fermat’s Theorem for a general n.

Since so many mathematicians of great renown attempted to solve

this problem and none of them ever solved it, for many years a prize of £5,000 was maintained for the person who solved this problem of

Fermat. Though many mathematicians terminated their pursuits of a general solution for this problem, due to the increased speculation that none existed, many continued to work on it due to the established reputation of Fermat. Among those mathematicians who continued to work on the problem was Rummer.

2 . Rummer and His Ideal Numbers

Gillispie[60] gives a brief biographical sketch of Rummer.

Ernst Eduard Rummer (1810-1893) was born in Sorau, Germany. After re­ ceiving his first formal instruction via a private tutor, Rummer entered the Gymnasium in Sorau in 1819 and the University of Halle in

1828. Originally Rummer was interested in Protestant theology. How­ ever, he gave up his theological studies in order to study mathematics under the influence of his mathematics professor, Heinrich Ferdinand

Scherk. In 1831, he was granted a doctorate in mathematics. From 1832 until 1842 he taught at the Gymnasium in Liegnitz, mainly mathematics and physics. During this period, among his many students were Rronecker and Joachimstal, both of whom became interested in mathematics due to 29

Kunmier. Also during this period, Kummer established himself in the mathematical world through his published mathamtical papers. In 1842, upon recommendations by Dirichlet and Jacobi, he was appointed a full professor at the University of Breslau. Both Dirichlet and Jacobi were greatly Impressed with Kummer's papers. Kummer was challenged by

Fermat's Theorem. Dickson[38] reports that Kummer devoted twenty of his best years to the study of this problem and related problems. His results were perhaps the most important contributions to this subject.

Kummer[116, 122], as well as several of his contemporaries, investi­ gated the equation

xP+yP = zP, where p is an odd prime. His rationale for this attempt was the fact that a proof for all primes p could imply a proof for any n.

Kummer observed that the equation xP+yP = zP can be written in the form

(x+y)(x4oy)(x+a^y) . . . (x+aP“^y) - zP, where a is a complex pth root of unity. (Kummer[119], as we shall later see, made a complete study of fields formed by adjoining the nth roots of unity to Q.) We shall illustrate the above decomposition for p = 3.

We know by De Moivre's Theorem that the cube roots of unity are

1, ” 1. +i(^),-l. -iC^),where i = Usually one sets 2 2 2 2

w - - Y + i (^ ) which implies that

= - I - i(^) . 30

Hence, as usual, the notation for the three complex roots of unity is

1, w, (j)Z; where = 1 and l+w+tijZ = 0. Thus

x^+yZ = g3 if and only if

( x 4y ) (x+wy) (x+ü)Zy) = because

(x+y) (x-hoy)(x+w^y) = if and only if

(xZ+tüxy+xy+üjyZ) (x+tü^y) = z^ if and only if

(x+w^y) (xZ+ojxy+xy+tüZy) = z^ if and only if

x^+ïi)xZy+xZy+ü)xyZ+(üZxZy+to^xyZ+ü)ZxyZ+(jü^y^ = if and only if

x^+(l+a>+ü)Z)xZy+(l+ü)+ujZ)xyZ+y3 = z3 if and only if

x3+y3 = z3.

This new method of decomposition led mathematicians of that era to study complex numbers of the form

X 2 +aX2 +o(Zx^+ . . . +a^ where x^, X2 . • • x^ are integers and, of course a is a pth root of unity, p being an odd prime. We mention again that Gauss, Jacobi and

Eisenstein had already made extensive use of the rings of integers

Z[i] and Z[w] in the algebraic number fields Q[i] and Q[dj] respectively.

This fact is the primary reason why mathematicians of that era were 31 2 3 attracted to these numbers of the form x^+a x^+ . . .

+a^ ^x . P Gabriel Larne[126] attempted to prove that Fermat’s Theorem for

any odd exponent n is not satisfied by the complex numbers of the

form mentioned above} that is, x^+y*^ = is not satisfied by

2 X = x,+ax„+a x_+ . . . *hx^ ^x 1 2 3 P 2 y = yi+ay2 +ot y^+ . • • z = z^^+azg+a^z^f . . . +a^ . P In fact. Lame thought that he had a proof. He had assumed unique factorization for the ring of integers in a certain algebraic number

field. This assumption was unfounded as was pointed out by both

Liouville[141] and Kummer[118] in terms of counterexamples.

In order to exemplify some of the thinking of Lamé and others, suppose we consider the set of odd primes 3, 5, 7, 11, 13, 17, 19, 23,

. . . in the ring of integers Z[i] contained in the algebraic number

field Q[i]. We can form two classes: those congruent to 1 modulo 4 and those congruent to 3 modulo 4. The classes are thus

5, 13, 17, 29, 37, 41, 53, . . . and

3, 7, 11, 19, 23, 31, . . . .

Now we observe that

5 = (2+i)(2-i)

13 = (3+2i)(3-2i)

17 = (4+i)(4-i)

29 = (5+2i)(5-2i). 32

Though these numbers are primes in Z, they are no longer primes in

Z[i]. Euler[49] later proved that every prime of the form 4n+l can be expressed in the above form; that is,

4n+l = (a+bi)(a-bi) or

4n+l = a^+b^, where a and b are integers in Z. In the other class of primes, that

is, the primes of the form 4n+3, this kind of factoring is not possible

since the square of an integer in Z is either congruent to 0 or 1 modulo 4. (This was established and utilized in the solution of x2+y2 = z^.) Thus

a^+b^ = 0, 1, 2 (mod 4);

that is.

a^+b^ $ 3 (mod 4).

Therefore

3, 7, 11, 19, 23, 31, 43, . . . remain primes in the ring of integers Z[i] of the algebraic number

field Q[i]. It is then obvious that the divisibility properties of integers in Z might fail for integers in Z[i],

As was indicated earlier in our discussion, Larne thought that he had proved Fermat's Theorem by using the ring of integers of certain algebraic number fields. In particular, he considered quantities of the form x+y/mwhere x, y, m are rational integers. Such a quantity

is called an algebraic integer. (A more precise definition is found

in Appendix E.) It was observed by Kummer[118, 119] that the factor 33 theorem of arithmetic "breaks down" when applied to algebraic integers.

We state this theorem below for convenience.

THEOREM IX-2 (Factor Theorem). If p is a prime in Z and p|bcj where b, ceZ, then either p|b or p|c.

Mordell[158] illustrates how this theorem fails in an algebraic number field by considering the integers of the algebraic number field

, that is, the numbers of the form x+yi/^, where x, yeZ. First, however, the following definitions are pertinent to our discussion.

DEFINITION II-l. The algebraic integer x + y / ^ is divisible by a + b / ^ if

x+y/^ _ (x+y/^) (a-b/^) a+b/“5 " (a^+5b^) is an algebraic integer.

DEFINITION II-2. The algebraic integer, x + y / ^ is prime if it is not divisible by another algebraic integer a+b/^, where a + b / ^ is not a unit, unless a = ix and b = iy.

Suppose we consider 21. Note that

2 1 = 3.7 = (4+/^) (4-/^) = (1+2^5) (1-2^5).

Now suppose that 4+v^T is not a prime algebraic integer; that is,

4 + / ^ = (x4yt^P5) (a+b/^) implies that

= (x-y/^) (a-b/^) by complex conjugation, which implies that

21 = (x^+5y^)(a^+5b^), which implies that

x^+5y^ is a factor of 21 34 which implies that

x^+5y^ must be 1, 3, 7 or 21.

Now

x2+5y2 - 2 implies that

X = ±1, y = 0 and

x^+5y^ = 21 implies that

X = ±4, y = il or

X = il, y = ± 2 .

But neither x = il, y = 0, nor x = i4, y = il gives a non-trivial factor of : that is, 4+ / ^ is a prime algebraic integer. Further

1+2/^, 3 and 7 are prime algebraic integers. Hence 3 1 (4+/^) (4-^^) but 3^(4~y^) and 3/|^4-/^) . Thus the factor theorem of Z is not valid in Z[/^]. ' It was this defect more than others which led Kummer to invent his ideal numbers.

Kummer[1 2 1 » 123] devised a method to re-establish the fundamental laws of arithmetic not quite within the set of algebraic integers under consideration but in a larger set obtained by adjoining to a concrete set of algebraic integers certain abstract integers which Kummer called

"ideal numbers." We pause here to state formally some of the defini­ tions and properties pertinent to ideal numbers. 35

3. Some Fundamental 'Concepts of Ktunmer’s Ideal Numbers

Before stating the definitions and concepts pertinent to ideal numbers, it is necessary to state and discuss some aspects of complex numbers. We do this for two reasons: 1) Kummer defined his ideal numbers in terms of these concepts and utilized them in most of his major works. 2) The notation of Kummer's works on ideal numbers is very cumbersome and, therefore, some advance experience with it will prove to be most helpful to the reader. (For more detailed information, proofs of theorems and proofs of remarks, the reader is referred to

Kummer[121, 123] and Smith[189].)

DEFINITION II-3. Let p be a prime number and let a be a root of the equation

then any expression of the form 2 13" 2 Fp(a) = aQ+a^a+a2a + . . . + Sp_2^ » where Sq, a^, 8 2 » . . . S p ^ denote real integers, is called a complex integral number.

Example : For p= 3, a^-1 = g implies that a^+a+1 = 0. This a- 1 implies that a = w and a = Thus f(w) = ag+a^w. (Note that since

(jij2 g allowing higher powers of m merely duplicates the values already determined.)

Remarks

1. Every rational and integral function of a can always be reduced to this form. 36

2. For each complex number a. given a particular p, the expression Fp(a) = aQ+a-j^a+a2 a^+ . . . +ap_2 a^~^ is unique.

DEFINITION II-4. The norm of F(a) is defined to be the real integer obtained by forming the product of all the p-1 values of F(a^) for i = 1, 2, . . . p-1 so that

N(F(a)) = N(F(a^)) = . . . = N(F(aP"^)) = F(a)*F(a^) . . .

F(aP**l) .

Example : For p = 3, N(F(w)) = N(F(w^) ) = (aQ+a^^u) (ag+a^w^) .

But (ag+a^m) (àQ+a^m^) = a^+aga^w^+aga^^m+a^m^ = ag-aga^+a|, where

= 1 and (jj+(jj^ = -1.

Remarks

1. Complex integral numbers for given p ’s and a's are amenable

to the arithmetical operations of addition, subtraction and multiplica­

tion.

2. The division of complex integral numbers is accomplished by

transforming the divisor into the norm of the divisor, thus reducing

the divisor to a real number; that is,

f(a) = f(a)F(a^) F(a^) . . . F(aP~^) F(a) NCF(a))

?nd further f(a) is said to be divisible by F (a) when every coefficient

in the product

f(a)F(a^)F(a^) . . . F(aP“^)

developed and reduced in the form f(a) = bp+bj^a+b2 a^+ . . . -fbp^gaP"^

is divisible by N(F(a)).

3. If we consider the numbers ag, a2^, . . . &p„2 the

expression F (a) = * * * ^^p-2^^"^ as indeterminates, then 37 the norm of F(a) is a particular homogeneous function of order p-1 and of p-1 indeterminates. (We shall see in Chapter III that this approach was developed by Kronecker.)

4. In view of Remark 3, the question of whether or not a given real number is resolvable into the product of p-1 conjugate complex factors (see Appendix E) is the same as asking whether or not it is capable of representation by a certain homogeneous form which is, in fact, the resultant (see Chapter III) of the two forms

aQxP“^+aixP~3y+ . . , ap_2yP“^ ____

xP”^4ocP“^y+xP”^y^+ . . . +yP"^.

Kummer[119] concerned himself with the former statement of the above question while Dirichlet[41] was involved with the latter form of the question which involved the theory of forms and their resultants.

Though forms and their resultants will be discussed formally in

Chapter III, for the time being we can think of a form as a particular polynomial function and the resultant of several forms as being a particular factor of these polynomials.

DEFINITION II-5. A complex unit is a complex integral number whose norm is unity. Smith[189] cites the following facts about complex units which we state as remarks.

Remarks

1. If p = 3, there is only a finite number (six) of these units given by the formula

U = +a^. 38

Now since for p = 3, a = u; then for k = 0, 1, 2 we have that U = tl,

±ti), respectively, that is, 1, -1, w, -w, are the units.

2. If p > 3, the number of units is infinite.

3. It is always possible to assign a system of p-1 units, where

" = I (P-1) such that all units are given by the formula , n n n

where u , u , . . . u are the assigned units and k, n,, n„, . . . 1 2 P-1 ^ ^ n are real integers. p-1

DEFINITION II-6. A fundamental system of units is a system of units such that each unit of a particular set of numbers can be represented by products of elements (units) of this system times roots of unity.

Remarks

1. If p = 5, a^+a”^ is the only fundamental unit, so that every unit is given by the formula U = ta^(a+a-^)^.

2. If p = 7, the complex units are given by the formula

U = ta^(a^+a"^) ^(a^+a~^)

3. If p > 5, the number of systems of fundamental units is infinite.

DEFINITION II-7. Suppose d is the least positive integer such that

a^ = l(mod m); then a is said to belong to d modulo m . 39

Example : 4^ = 64 = l(mod 7) while 4^ = 16 = 2(mod 7). Thus 4 belongs to 3 modulo 7.

DEFINITION 11-8. An integer a that belongs to (f»(m) modulo m

(see Appendix E) is called a primitive root of m.

Example; (4) = 2. Thus = 3^ = l(mod 2^) but 3 t l(mod 2^) .

Therefore, 3 is a primitive root of 2^.

Remark

In group-theoretic language,a is a primitive root modulo p if a generates the group of units (see Appendix A). mZ

DEFINITION II-9. The e periods of f roots each are the quantities bg, bj^, b2 » . • • b^_^ defined by the equations

bg = a'^°+ad%a‘^^"'+ . . . +

bi = . . . a" " ' " " ' '

e-1 2e-l 3e—1 fe—1 b_ . = a° +a^ +ad + . . . a^ , e-1 ' where d denotes a primitive root of p. These are the roots of an irreducible equation of order e having integral coefficients, where

ef = p-1 for the particular prime p. We shall symbolize this equation by

F(y) = y®+A2 y®“^+A2 y®”^+ . . . +A^_^y+Ag = 0.

Remarks

1. The determination of the coefficients of the equation F(y) =

0 above may be accomplished for any prime p and any divisor e of p-1 40

by methods that are very tedious though not theoretically difficult

(see Gauss[56]).

2. Each of the e periods is a rational function of another

period in such a way the one can establish the equations

bj^ = f|)Cbo)

b2 = ({«(b^)

^e = where it is to be observed that the coefficients of the function (}) are

not, in general, integral.

3. Every rational and integral function of the periods can be

reduced to the form

agbo+aj^bj^+a2b^+ . . .+ae-l))e-l'

The following theorem expresses an arithmetical property of the

equation F(y) = 0 which is very significant in the theory of complex numbers.

THEOREM II-3. If q is a prime number satisfying the congruence

q^ = 1 (mod p),

then the congruence

F(y) = O(mod q)

is completely factorable; that is, it is possible to establish an

indeterminate congruence of the form

F(y) = (y-Xg) (y-Xj^) . . . (y-x^_j^) (mod q),

xg, X]^, . . . Xg_i denoting integral numbers determined mod q. 41

A profound relation exists between the periods bg, b^, . . r bg.^ of the equation F(y) = 0 and the roots Xg, x^, . . . x^_^ of the con­ gruence F(y) E O(mod q). In particular, this relation can be observed in the following theorem:

THEOREM II-4. Every equation which exists between any two functions of the periods will exist as a congruence for the modulus q when we substitute for the periods the roots of the congruence F(y) =

O(mod q) taken in a particular order.

We return to the complex integral numbers of the form F (a) = ag+ a^a+a2 a^+ . . . fa^^ga^"^ in order to cite more of their division properties. Suppose we can replace the complex number f(a) = ag+a^^af a2 a^+ . . by the complex number g(bg) = Cgbg+Cj^bj^+C2 b 2 +

. . . which, with its conjugates

g(bj^) = Cgb'j^+Cj^b2+C2b3+ . . . +Cg_]^b^

g(6 2 ) = Cgb2 +C]^bg+C2 b^+ ■ . .

“ S^e-l‘^l^O'^^2’^1’^ ' ' ' '^*^e-1^2e-2• is a function of the periods only and is therefore a special form among the general complex numbers f(a). Now suppose we let q be a real prime satisfying the congruence

q^ = l(mod p).

Kummer[119], by means of the relation existing between the roots of the equation bg, bj^, b2 , • . . b^^^ and the roots of the congruence 42

Xq , established the two theorems which follow;

THEOREM II-5. The necessary and sufficient condition that gCbg) be divisible by a, where q is a prime belonging to the exponent f, is that the e congruences

eCxq) “ CqXq+Cj^x-j^+C2 X2 ^" . • • ^Cg_^x^ _ 2 — O(mod q)

g(%l) = CqXj^+C2 X 2 +C2 X3 + . . . +C2 _^Xg_i = OCmod q)

B(Xe-l) = CoXe_i+CiXo+C2%l + ' ' ' ^e-1^2e-2 = are simultaneously satisfied. (Note that this divisibility is in terms of complex integral numbers.)

THEOREM II“6. The necessary and sufficient condition that the norm of gCbg), taken with respect to the periodsj that is, the number g(bQ)g(b^> . . . g(bg_^), is divisible by q, where q belongs to the exponent f mod p, is that one of the e congruences

gCxg) = O(raod q)

g(x^) = O(mod q)

g(%e_l) = O(mod q) is satisfied.

Remark

The two theorems just cited above may be extended to any complex number f(a) by reducing it to the form 43

f(a) = gQ(bQ)+ag^(bQ)+a^g2 CbQ)+ . . . a^'lg^^^Cbg).

The definition of a prime q belonging to an exponent f is characterized by the congruence

q^ = iCraod p) such that q does not satisfy a congruence of lower index and of the same form. In view of this. Theorems II-5 and II-6 may be replaced by two analogous theorems which actually include them.

THEOREM II-5’. The necessary and sufficient condition that f(a) is divisible by q where = l(mod p) is that the congruences

Sq Cxj^) = O(mod q)

glCxfc) = O(mod q)

gf_l(xj.) 5 O(mod q) are simultaneously satisfied for every value of k. (The division is complex integral number division.)

THEOREM II-6’. The necessary and sufficient condition that the norm of f(a) be divisible by q, where q is a prime satisfying the congruence q^ 5 (mod p), is that the congruences

go(xjj.) = O(mod q)

gj^(xj^) = O(raod q)

Bf-l(*k) = q) are satisfied for some one value of k. 44

DEFINITION 11-10. f(a) is said to be congruent to zero (mod q) for the substitution b^ = x^, when the congruences

gQ(Xj^) = O(mod q)

gl(Xj^) = O(mod q)

g^_^(x^) = O(raod q) are simultaneously satisfied.

Remarks

1. If we denote by FCb^) the complex number involving only the periods which we can obtain by multiplying together the f complex numbers

f(a), f(adS), f(ad^®), . . . f(ad^^"^^®), it can be proved that the single congruence

F(x^) 5 O(mod q) is precisely equivalent to the f congruences gQ(x^) = O(mod q), B]^(Xj^) =

O(mod q), . . . gf_j^(xj^^) = O(mod q).

2. If we denote by G(bg) a complex integral number congruent to zero for every one of the substitutions = *1 * bg = Xg, . . . bg = x^_^, but not congruent to zero for the substitution b^ = Xq, it can be proved that the f congruences gg(x^) = O(mod q), g^(x^^ = O(mod q) . . . gf_l(Xj^) = O(mod q) are equivalent to the single congruence

G(gg_k)f(a) 5 O(mod q).

3. The single congruence of Remark 1, F(x^) = O(mod q), is very

Important because it demonstrates the important proposition: If 45 gQ(bo)Si(bo) = O(mod q) for bg = x^, then either gQCbg) = O(mod q) for bg = x^ or giCbg) = O(mod q) for bg = x^.

Now that the basic principles of Rummer's complex number theory have been established,we can proceed to discuss his ideal numbers. One will soon observe that the theory of Rummer's ideal numbers is rather vague in spite of the fact that he utilizes most of the preceding more concrete results.

Suppose q is a prime belonging to the exponent f,where ef = p-1.

Further, suppose that q can be expressed as the norm taken with respect to the periods of a complex number g(bg) which contains the periods of f terms only so that

q = g(bg)g(b|^) . . . g(be_i).

Now if g(Ug) 5 O(mod q) for the substitution of Ug in g we can dis­ tinguish the e factors of q by means of the substitutions which respectively make them congruent to zero.

THEOREM II-7. Suppose q^ is a factor of q such that q-^ belongs to xg modulo q. If f(a) is congruent to zero modulo q for any sub­ stitution bg = Xg, then f(a) is divisible by q^.

Remarks

1. If g(bg) is a factor of q, then

f(a) ^ f(a)g(bi)g(b2 ) • . . g(be-i.) g(bg) q ’ but since f Ca)g(b]^)g(b2 ) . . . g(bg_^) is congruent to zero modulo q, for every one of the substitutions bg = Xg, bg = x^, bg = Xg, . . . bg = x^it is divisible by q; that is, f(a) is divisible by g(bg). 46

2, It can be shown that the complex factors of q are primes in the proper sense of the word; that is, first that they are Incapable of being factored into any two complex integral factors unless one of these factors is a complex unit and secondly if any one of them divides the product of two factors, it necessarily divides one or the other of the two factors separately, (Here we are assuming that p remains con­ stant and that the division is complex integral number division.)

DEFINITION 11-11. Let q be any prime number belonging to the exponent f and let f(a) denote a complex number which is congruent to zero modulo q by the substitution of bg = Xg; f(a) is said to contain the ideal factor of q belonging to the substitution bg = Xq .

Remarks

1. To obtain a definition of the multiplicity of an ideal number

(factor) we can employ the complex number g ( b g ) which we introduced earlier.

2. If [g(bg)]”f(a) = O(mod q") is satisfied but I g ( b g ) (a) =

O(mod q^"*"^) is not satisfied, then f(a) is said to contain n times the ideal factor of q which belongs to the substitution bg = Xg.

Kummer[119] introduced several theorems which characterize his ideal factors (numbers). We state some of these theorems here, without proof, in our continuing effort to clear-up some of the vagueness surrounding these numbers.

THEOREM II-8. A complex number is divisible by q when it contains all the ideal factors of q. If it contains all of those factors n times but not all of them n+1 times, it is divisible by q^ but not 47

THEOREM II-9. The norm of a complex number is divisible by q when the complex number contains one of the ideal factors of q. If,

counting multiple factors, it contains, in all, k of the ideal factors

of q, the norm is divisible by q^^, but by no higher power of q, where

f denotes the exponent to which q belongs.

THEOREM 11-10. A product of two or more factors contains the

same ideal divisors as its factors taken together.

THEOREM 11-11. The necessary and sufficient condition that one

complex number is divisible by another is that the dividend contains all of the ideal factors of the divisor at least as often as the divisor does.

THEOREM 11-12. Two complex numbers which contain the same ideal

factors are identical, or else differ only by a unit factor.

THEOREM 11-13. Every complex number contains a finite number of ideal prime factors.

Remarks

1. The number of ideal prime factors into which a given real prime can be factored depends on the exponent to which the prime belongs for the modulus p.

2. If the exponent to which a prime p belongs is f, then the total number of ideal factors is

= e.

3. If q is a primitive root of p, then q continues to be a prime in this complex number theory; if it is a primitive root of the congruence 48

27(P"^) = Krnod P),

it is only factorable into two conjugate prime factors,

4, Every ideal factor is a divisor of an actual (complex) number,

and indeed, of an infinite number of actual (real) numbers.

5. If the ideal number i|)(a) is a divisor of the actual number

F(a), then the quotient

(|3^(a) = F(a) ?

is always ideal.

Before we state more results pertinent to Kumraer's ideal numbers, we shall give a concrete example. The following example is due to

Dedekind[33].

Example: Suppose we consider the quadratic field (algebraic number field), Q [ / ^ ] . Let R be the ring of integers in Q [ / ^ ] , that

is, the set of elements of the form x+y/^, where x and y are rational

integers (see Appendix E), Suppose w = x + y v ^ is an integer in Q[/^].

Then the norm of w

N(w) = (x+y/^) (x-y/^) = x^+5y^ and hence the only unit elements of R are the numbers il due to the

theory of Pell's Equations (see Appendix E).

Observe that in R

21 = 3.7 = (1+2/^) (1-2»/^) ,

that is, 3*7 = (1+2/-5) (1-2/^) . Thus factorization into primes in R

is not unique. Kuraraer[119] then introduced four ideal factors a^,

g2 such that 49

3 =

7 = SjBj

1 + 2 » ^ = and

1 - 2 / ^ =

(Note that a^, «2 * 6^, ^ 2 sre not elements of a particular algebraic number field but merely Inventions, as we shall see, used to recapture the law of unique factorization.) Now an arbitrary integer of R w = 6t+y/^)is divisible by Oj, Og* ^ 2 the following congruences are satisfied:

(1 -2 / ^ ) (x+y/^) = O(mod 3)

(1 + 2 / ^ ) (x+y/^) = O(mod 3)

(l+2/=5) (x+y/^) = O(mod 7)

(1-2/^) (x+y^5) = O(mod 7).

But (1 -2 /^) (x4-y/^) = (x+lOy) + (y-2x)/^ and (1+2/^) (x4y/^) =

(x-lOy) + (y+2x) Hence the four congruences above reduce to

x+y = O(mod 3)

x-y = O(mod 3)

x-3y 5 O(mod 7)

x+3y = O(mod 7).

Further, let w^ = Xj^+y-j^/^ be another arbitrary integer in R. Then

WW]^ =(x+y/^) (x^+y^/^) = X2 +y2 *'^» where X2 = xx^-Syy^ and

5^2 ” Hence we can deduce the fact that

X2 +y2 = (x+y)(x^+y^)(mod 3).

But since (1-2*'^) (x+y/^) = O(mod 3), (1+2/^) (x+y/^) - O(raod 3)

(l+2v'^) (x4y/3) = O(mod 7) and (1-2/^) (x4y/^) = O(mod 7), then wWj^ is 50 divisible by if and only if one of the two factors w or w^ is divisible by ww^ is divisible by if and only if w or w^ is divisible by a^. Similar results hold for and Hence we have, in view of these results, that 1 + 2 is divisible by but not divisible by « 2 and 1 - 2 / ^ is divisible by and not by Thus and are two distinct prime ideal factors. Utilizing our congruences again, we can deduce the fact that an integer w = x + y / ^ is divisible by 3 if and only if it is divisible by as well as «2 " This was Rummer's justi­ fication for setting 3 = and similar arguments justify 7 = g^gg,

1 + 2 / ^ = 1 -2 / ^ = Cgg^. Thus, regardless of the factorization,

2 1 = by introducing these and other ideal numbers, Rummer was able to recapture unique factorization in algebraic number fields.

An ideal number, as we have demonstrated, is not capable of being exhibited in an isolated form as a complex integral number. In other words, an ideal number has no quantitative existence. When Rummer said that a given complex number contains an ideal factor, this was only a convenient mode of expressing a certain set of congruence conditions which are satisfied by the coefficients of certain complex numbers. In spite of these facts. Rummer[119] proceeded to classify his ideal numbers.

Referring to ordinary real numbers as actual numbers, Rummer asserted that every ideal number was a divisor of an ordinary number and that the quotient of such division is necessarily ideal; that is. 51 if F(a) is an actual number such that the ideal number ij>(a) divides

F(a), then

4»!(a) = F(a) f 4>(a) is an ideal number. He then asserted that ideal numbers are distributed into a certain finite number of classes, where a class consists of those numbers which become actual numbers when multiplied by the same multiplier (factor). Thus ideal numbers which belong to the same class are said to be equivalent. The set of actual numbers form the first or principal class since they do not require an ideal multiplier.

Rummer was successful in determining the number of non-equivalent classes of ideal numbers.

Rummer[117] determined the number of non-equivalent classes of ideal numbers to be the number H defined by

“ ° - !• where P is defined by the equations

P = (B) *

*(3) = l+d3^g+d2 g^+d3 6 ^+ . . . dp_2g^ , where 6 represents a primitive root of the equation

= 1, d a primitive root of the congruence

dP“^ = Krnod p), dj^, d 2 » dg, . . ., the least positive residues (see Appendix E) of 9 o d, d^, d for modulus p. û is the logarithmic determinant of any system of p-1 fundamental units (see Appendix E) defined by 52

LCj^(a), LC2(s ) , •

LCj^Ca*^), LC2(a'^).

p-2 U-2 p-2) LC^(a(l ), LCgCa'l ), . . . LCp_^(a‘^ where d is a primitive root of p, G^(a), € 2 (3 ) Cp_i(a) is a system of independent units (see appendix E), p = and a is still a root of the equation

and D is the logarithmic determinant of a particular system of inde­ pendent but not functional units e(a), e(a'^), e(ad^), . . , e(a^^ defined by the equation

/(1-a^)(l-a"^y + ad^^-i>(l-ad) ^ sin kd e(a) V (1-a) (1-a-l) = - ÏII " - ____ p , if a = e so that -2 Le(a), Le(a^), L(e(a^ Le(a^^ ) j j2 j3 jW-1 Le(a ), Le(a*^ ), Le(a° ), . . . Le(a“ )

A = Le(a^^), Le(a'^^), 1 0 (3 ^^ ), . . . le(a'^^)

.dP, d2y-4, Le(a^ , Le(a^^ , Le(a^^), . . . Le(a

Kummer[115] was able to show that both of the two factors

E— and 2. are integers. The factor 2 represents the number of (2p)W-l 4 A 53 classes that contain ideal numbers composed with the periods of two terms a+a”^, a^+a~^, . . . only. On the other hand p represents (2p)P“^ the number of these ideal numbers which become actual by multiplication with their own reciprocals. For the cases where p = 5 and p = 7, the trigonometric units e(a), eCa*^), e(a*^^), . . . are themselves a funda­ mental system so that in these two cases D = A, and thus JD = il. A

The number H of classes of ideal numbers in general is not divisible by p. However, there are certain cases where it may happen to be. The quotient D is never divisible by p except when the other quotient A is also divisible by p. Kummer[116] found that the necessary and sufficient condition for the divisibility of ---?--- by p is that the numerator of the first w-1 functions of Bernoulli (Bernoulli

Numbers) is divisible by p. Kummer called those primes p which divide

H regular primes. We shall discuss these primes in more detail in

Section 8 of this chapter.

Though Rummer's ideal numbers are somewhat versatile in that their adjunction recaptures some of the fundamental properties of arithmetic, they seem to be rather vague and not clearly defined. For this reason most mathematicians abandoned Rummer's theories on complex (algebraic) numbers in pursuit of proofs or disproofs for their own theories and conjectures. However, since Kummer invented these numbers in his attempt to solve the Fermat Problem, he appeared to be content with this vagueness. 54

4, Application of Ideal Numbers to Fermat's Problem

Kummer[116] showed that for complex (algebraic) numbers of the form 2 a^+agO+agO + . . . + there are corresponding ideal numbers of the form

------7 a^+a2 a+a3 a2 + . . . are integers satisfying the where a^^, Sg, . . . are integers satisfying the conditions out­ lined in Section 3 and is a factor of the number of classes of ideal numbers. We know from Section 3 that R depends on p, where p is an odd prime. Returning to the original decomposition of the equation xPdyP = zP; we recall that

xP+yP = (x+y)(x+ay)(x+a^y) . . . (x+aP”^y) =>

(x4y) (x+ay) (x+a^y) . . . (x+aP~^y) = zP.

Now suppose z and p are relatively prime. This meansthat the left hand side of the equation is such that no two factors havea factor in common. By introducing ideal numbers we obtain

x+ay = yg(bg) ]^

x+a^y = C^gi(bg) ]P

x+a^y = where g(bg), g^(bg), . . . are ideal numbers and ^2 * ^3 * * * ' units and can be expressed in the form 2 x^ +Xg a+x^ a + . . . .+.

Now we know that g(bg) is of the form which we indicated earlier, that is, 55 H______gCbg) = .

so we have H______x+ay = ! i ) .

Now if (H, p) = 1, then we have that * • • is the Hth

power of an expression of the same form. Hence

x-fay ° ^2 (X]^+X2 a+X3 a^+ . . . +)P,

where x^, X9 , . . . are integers. Similarly we can find corresponding

equations for x+o^y = ^[g(bg)]P, x+a^y = Sg[g(bg)]P, .... Kummer

[116] showed that these equations (x+ay)= (x2 +X2 a+Xga^+ . . .+)P,

etc, are impossible not only when x and y are ordinary integers but p also when they are integers of the form x^+Xga+x^a + . . . . One can

also observe that Kummer similarly drew such a conclusion when z and p

are not relatively prime. Hence Kummer proved Fermat's Last Theorem

in all casés where H is relatively prime to p.

Though the invention of Ideal numbers was most significant to the

development of ideals, as we shall see‘later, ideal numbers as a subject

themselves were not studied very much by other mathematicians in

subsequent years.

5. A Later Development Concerning Ideal Numbers

^ Kummer [121] observed that an analogy existed between his ideal

numbers and chemistry. Hancock[75] has summarized this analogy with

some excellent examples due to Sommer[190] and Hensel[81]. We shall

illustrate these examples before we discuss the actual analogy and its

ramifications. 56

Suppose we consider the set of natural numbers

N = {1, 2, 3, 4, 5, 6 , 7, 8 , . .

We can now form, from this set, two sets Cq and by the following;

1. Let Cq be composed of those natural numbers which have an even number of prime factors.

2. Let be composed of those natural numbers which have an odd number of prime factors.

Thus

Cq = (1, 4, 6 , 9, 10, 14, 15, 16, 21, 22, 24. . .} and

= {2, 3, 5, 7, 8 , 11, 12, 13, 17, 18, 19, 20, 23 . .

Now suppose we consider only the set Cg as a fixed set of integers.

Observe that

210 = 6*35 = 10*21 = 14*15 which reveals the fact that 2 1 0 can be factored as a product of

"prime" integers in Cq in three different ways. We can write

210 = (6,4) (6,21) (35,10)05,21) = 2'3'5'7

210 = (6 , 14)(6,15)(35,14)(35,15) = 2*3*7*5

210 ■= (10,14) (10,15) (21,14) (21,15) = 2*5*7*3.

We can see that several of the factors are equal; namely

(6 ,1 0 ) =(10,16) = (10,14)

(6,21) = (6,15) = (21,15)

(35,21) = (35,14) = (21,14)

(35,10) = (35,15) = (10,15).

We can designate Cq to be the set of actual integers and Cj^ to be the set of ideal integers. It is interesting to note that the integer 210 57

can be expressed as the unique product of four ideal integers 2, 3, 5,

7 though these integers do not exist in Cg. We note that this

factorization is similar to the factorization of 2 1 in the algebraic

number field Q(/^) which we discussed in Section 3, that is,

21 = 3-7 = (l+2/:5)(l-2/:5) = (4+v^) (4 - ^ ) .

Another example, used by Sommer[190], follows below:

Suppose we consider the set of integers of the form 4n+l as a

fixed set; that is,

S = {1, 5. 9. 13, 17, 21, 25, 29, . . ., 45...... 117, . . .}

Suppose a, beS where a = 4kj^+l and b = 4 k 2 +l. Then observe that

ab = C4kj^+1) (4 k2 +l) =>

ab = 1 6 k^k2 4 4 k^+4 k 2 +l =>

ab = 4 (4 k]^k2 + k K k 2 ) +1-

Now 4 kj^k2 +k^+k2 is an integer. Denote this integer by k. Hence ab is

of the form 4k+l which implies that abeS. Thus a, beS implies that

abeS. The integers 5, 9, 13, 17, 21, 29, . , . are therefore the prime integers of S since they are irreducible in S. We now consider

the integer 10,857 = 4(2714)+1eS. Observe that

10,857 = 141*77 = 21*517, where 21, 77, 141 and 517 are prime integers in our fixed set S. The

fact that 10,857 = 141*77 = 21*517 is proof that the factorization of

an integer into its prime factors in S is not unique.

Since the factors 3, 7, 11, 47 do not exist in S, Kummer would

replace them by ideal numbers. Set 7 = (21,77); 11 = (517,77); 47 =

(517,141); 141 = (9,21)(141,517); 77 = (77,21)(77,517) and observe

that 58

10.857 = (141,21)(141,517)(77,21)(77,517) = 141*77.

Also

10.857 = (21,14)(21,77)(517,77)(517,141) = 21*517.

In both cases, the decomposition of 10,857 into its ideal prime

factors is unique. As was demonstrated in Section 3 of this chapter,

Kummer invented his rather vague "ideal" numbers in order to achieve unique factorization. Returning to the idea of an analogy between chemistry and ideal numbers we summarize as follows:

1. The chemical elements correspond to prime factors.

2. The multiplication of complex (algebraic) numbers corresponds to the chemical compound. Example: Let 4 + / ^ correspond to Na and

4 - / ^ correspond to Cl, then (4+/^) (4-/^) = 21 corresponds to NaCl.

3. The chemical formulae for the decomposition of compounds correspond to the factorization of numbers. Example : NaCl -*-Na+Cl and

21 = (4+v^) (4-/^) .

4. Kummer's ideal numbers correspond to the hypothetical radicals which have not been isolated but exist in combinations with other elements.

5. The concept of equivalence in chemistry corresponds to the concept of equivalence in the theory of complex numbers. Example: In chemistry two weights are equivalent if they can replace each other

(they are isomorphic) for the purpose of neutralization; in our number

\ theory two ideals are equivalent if each of them can make a rational number out of the same ideal number.

The analogies above illustrate the fact that vague concepts may have some rather precise analogies. Some of this vagueness of 59

Kummer's ideal numbers, however, will disappear when Dedekind replaces them by his ideals.

6 , Dedekind and His Ideal Theory

Richard Dedekind was one of Kummer's most able followers.

Gillispie[60] is the basic reference for the brief biographical sketch which follows: Richard Dedekind (1831-1916) was born in Brunswick,

Germany. Between the ages of seven and sixteen, he attended the

Gymnasium Martino-Catharinem in Brunswick. Though Dedekind was initially interested in chemistry and physics and considered mathe­ matics as an auxiliary science, he later became interested in mathe­ matics while attending Gottingen.

At Gottingen, Dedekind developed a close friendship with Bernhard

Riemann, who also was a student there during this period. It was this friendship which drew him closer to mathematics. During the summer of

1854, he, along with Riemann, qualified as a university lecturer in mathematics. During the winter semester of the 1854-55 Jaeademic y e a r , hé began his teaching activities as !Privatdozent. After Dirichlet succeeded Gauss at Gottingen in 1855, Dedekind attended his lectures on the theory of numbers, potential theory, definite integrals and partial differential equations. He later entered into a close, per­ sonal relationship with Dirichlet and had many fruitful discussions with him. One can see that Dedekind had studied several mathematical topics of his time before he became interested in the theory of generalized complex numbers. 60

During the latter part of the 19th century, Dedekind, Kronecker, and Weierstrass became involved with the editing of the mathematical works of their former colleagues and deceased teachers whom they held in highest esteem. Dedekind, while editing some of Dirichlet's lecture notes, became seriously interested in the theory of generalized complex numbers and their decomposition into linear factors. In 1871,

Dedekind published these lecture notes of Dirichlet, along with a supplement in which he established a theory of algebraic number fields by introducing ideals.

Dedekind[33] defined his ideals in a more explicit manner than

Kummer had defined his ideal numbers. We shall later observe that though Dedekind worked extensively with commutative rings and fields, he defined his ideals (divisors) in terms of the numbers in the partic­ ular ring under consideration. This made ideals much more tangible than the abstract ideal numbers. Dedekind also differed from Kummer in that his methods were applicable to all^ algebraic number fields rather than those defined by roots of unity only. For this reason.

Dedekind is recognized as the founder of modern multiplicative ideal theory. It is significant to note that from the very outset Dedekind was interested in the structure of the generalized complex numbers themselves rather than in their application to Fermat's Problem.

Dedekind[34] observed that a finite field extension L over Q, the field of rational numbers, is either an algebraic number field or an algebraic function field with one variable (see appendix A). Dedekind worked, for the most part, with the field of rational numbers as his 61

ground field because this field has the properties which characterize

fields later called Dedekind fields. We give a formal definition of a

Dedekind field.

DEFINITION 11-12. The number a in k is said to be an integer in

k if it satisfies an equation

f(a) = a^+aj^a^“^+ . . . +a^ = 0 , where a-^, S2 , a^, . . . a^sZ; 1 , 2, 3, . . . h-1, heZ"^ (positive

integers), and k is an extension of Q.

DEFINITION 11-13. Suppose k is a field such that whenever the

Integers a^, 3.2 , . . . a^ are without a non-trivial greatest common

divisor (see Appendix E), that is,

(a-j^, ag, a^ . . . a^) = (1)',

then there exist integers such that

^1 ^1 '^^2 ^2 '^ . . . ia^Ak “

Then k is called a Dedekind field.

Remarks

1. In more modern terminology this definition is equivalent to

saying that k is a field in which every ideal (divisor) in its ring of

integers is the product of prime ideals (divisors).

2. The terra divisor was used to Indicate what we call an ideal

today.

In view of the above property of the field of rational numbers,

Dedekind gave these definitions for a divisor (ideal).

DEFINITION 11-14. Any finite sequence of integers aj^, 3 3 , . . .

a^ in L determines a divisor a; the integer a is said to be divisible 62

by if there exist integers ^2 * * * ' ^n that

a = a^J.j^+ajÎ2 + . . .

DEFINITION 11-15. A divisor is a non-empty subset of the ring

of integers R of a field k having these two properties;

1 . a±b belong to a if a and b do.

2. Xa belongs to a if A is any element of the ring R and a belongs to a.

The term divisor as a forerunner for the term ideal originated as

follows; Dedekind[33] observed that a set of integers S in the ring of

integers R of a field such that each integer in S is divisible by an

integer d has the properties: 1) If a, beS then a±beS, that is, a±b

is divisible by d. 2) If aeS and AeR, then AasS, that is, Aa is

divisible by d. Thus S would be referred to as a divisor and could be

a replacement for d.

Example; In the ring of integers Z, the integer 3 could be

replaced by the divisor

(3) = {. . . -15, -12, -9, -6 , -3, 0, +3, + 6 , +9, . . .}.

Remarks

1. Dedekind would refer, to 9, for example, as an element of the

divisor (3) rather than saying that 9 is divisible by 3.

2. The divisor (3) = [A3|AcZ) is a particular kind of divisor

called a principal divisor (ideal).

3. Since Dedekind primarily worked with the field of rational numbers and its ring of integers, he often worked with principal

divisors (ideals). 63

Example; Suppose S is the set of all integers of the form a+b/^, where a and b are integers in Z; that is, S = Z ( / ^ ) . Observe that

(a+b/^) Ca-bv'^) = a^-b^(-5) implies that

Ca+b/=5)(a-b/=5) = a^+5b^ 2 2 but a +5b is an ordinary integer. Thus by our previous work, we see that any divisor of 5 contains integers. Let d be a divisor of S, then d of course contains rational integers. Let k be the greatest common divisor of the rational integers in d: hence k is a linear combination of these integers. This implies that k|d since d is a divisor of S.

Similarly if we set m equal to the greatest common divisor of d, then d contains & + m / ? for some integer d then contains [k, A+m/^], where

[k, A+m/^] is the set of all homogeneous linear combinations of k and

A+m / ^ with integer coefficients. Further, according to Dickson[38], d = [k, A+m/^],

Let a+b/^ed. If b = 0, then a is a multiple of k. If b 0, then b == qm for some integer q. Hence d contains

a+b/^5-q (A+m/^) = a+b/^-qA-qm/^ so that

a+b q ( A+m/^^ = a-qA.

But since a-qA is a rational integer, then it is a multiple of K. Fur­ ther, since [k, A+m/^] d, then it is true that if a + b / ^ is an element of d and k is an element of Z, then k(a+b/^) must be an element of d.

We next observe what happens to k / ^ and (A+m/^)/^; that is, are k / ^ and - 5 m + / ^ elements of d? 64

The coefficients k and A must be multiples of m. Let k = am and

A ■“ vbm. Now observe that

k / ^ = abm+k/-3-abm

» a(bm)+k/^-b(am)

= aA+am/^-bk

= a(A+m/^)-bkcd.

Also

(A+m/^) = “5m+A/^

= -5 m+A/-5+b ^m-b ^m

= b*bra+bm/^-m(b^+5)

= b(A+m/^)-m(b“+5) .

But

b(A+m/^)-m(b^+5)ed <=> k|m(b^+5) .

Therefore

d = [am, m(b+/^) ], where b^ = -5(mod a). So we see that d is the totality of all linear

(homogeneous) combinations of k and A + m / ^ with integer coefficients.

From our discussions above, we see how Dedekind defined his divisors.

Now we should like to see just how he operated with them.

7. Some Basic Operations on Ideals

Dedekind[34, 35,.36] used his ideals in the same manner that most mathematicians used integers in 2. We shall state and, in some cases, prove several theorems which will demonstrate some of Dedekind's methods and various properties of Dedekind fields. Suppose k is a

Dedekind field. 65

THEOREM II-14. If a is divisible by the divisor

a = (hjl, 3 2 , . . • aj^), then there exist integers A2 , . . . A^ek such that

a = ’ * .da^A^.

Proof;

Suppose

B = (bi, b2, . . . bg) is a divisor such that ag = (b) is a principal ideal. Then every integer divisible by abj, for eachj, is divisible by b (written abjlb). Set

ab, — J- = c .. b 3

Then the greatest common divisor (gcd) of all integers n=i 2 3 b . . . r; j = 1, 2, . . . s) is 1. Hence we can obtain integers A^j such that

because we are working with a Dedekind field. Now

■ij implies that

a.ab.

but _ Hence . a . . «= a imp^es that b j ' b

i^j ai^ijCj = a. 66

Let =5 A^. Then ^Z^a^A^jCj = a implies that

i*i%i a; that is,

a =

THEOREM 11-15. A finite field extension L over a Dedekind field k is a Dedekind field.

THEOREM 11-16. Let k be a Dedekind field and let a be a divisor in the ring of integers of k; then the congruence

ax = bCmod a) is solvable for x if b:(a,a).

Proof ;

Suppose a ~ (a^,a2 » . . . a^) and b;(a,a). Since k is a

Dedekind field, there exist integers A^, A2 , . . .A^ such that

b = aA+a2 A-j^+a-jA2 +a3 A'^+ . . . +aj-A^, which says that

b = aA(mod a),

Thus

ax = aA(mod a), which implies that

X = A(mod a).

Hence ax = b(mod a) is solvable for x.

Corollary. If p is a prime in k and a jï O(mod p), then the congruence

ax = b(mod p) has a solution x which is uniquely determined mod p. (Hence the residue classes modulo a prime divisor a not only form a ring without zero divisors but also form a field.) 67

We leave the properties of a Dedekind field to focus our atten­ tion on some of the basic operations involving Dedekind's divisors.

Weyl[215] has summarized Dedekind's basic operations as follows:

Let k be a field. Some of its elements are called integers and certain classes of elements are called divisors. The basic relation in this field is "divisibility of" and is denoted by

Integers

1. The unit 1 is an integer.

2. The sum, difference and product of two integers are

integers.

3. Every element may be written as a fraction where a

and b are integers and b ^ 0 .

Divisibility

DEFINITION 11-16. Let a and g be divisors; then a;g means that every Integer a divisible by a is divisible by g.

1. Let a and b be integers and a a divisor. If a:a and i,

is an integer, then &a:a. If a:a and b:a, then

(atb):a.

2. If a:g and g:o, then a = g, where a and g are divisors.

3. There exists a divisor tt such that Ir-rr.

4. Every integer is divisible by ir, and hence r is unique.

Remarks

a. Statement 1 points out the fact that the elements

divisible by a given divisor form a divisor (ideal)

in the ring of integers of k. 68

b. Statement 4 demonstrates the fact that there

exists a unit divisor, ir.

Multiplication of Divisors

1 . OTT = a.

2. ag = ga.

3. (ag)v = aCgy) •

4. (ag) : a.

5. a:iT implies that a = tt,

6 . a;g implies that ayîgy.

Multiplication of a Divisor by An Integer

1. Let a ^ 0 be an integer and g be a divisor; then there

exists a divisor ag such that ab:ag if and only if b is

an integer divisible by g.

2. Let a # 0 be an integer and let a be a divisor such

that every integer c:a is such that c:a, then there

exists a divisor g such that a = ag,

3. a(ag) = (aa)g.

4. a:a is equivalent to atr:a.

5. a:air is equivalent to the statement that every integer c

:a is :a.

DEFINITION 11-17. air is called the principal ideal (a).

6 . a(by) = (ab)y, where a and b are integers and y is a

divisor.

Some More Explicit Statements

1. Let a be a divisor; then there exists a divisor a' such

that aa' is principal. 69

2 . ayîBy implies that a:g.

3. ay = gy implies that a = g.

4. If a:g then there exists a divisor y such that a = gy.

5. a:a, b:g implies that ab:ag.

DEFINITION 11-18. P is said to be a prime divisor if P = P is its only proper factorization, where P^Pg . . . P^ is a proper factorization of P if no one of the factors P^ equals ir.

6 . Given a divisor a, there is a natural number h such

that a allows no proper factorization into more than h

factors.

7. If P is a prime divisor and ag;P, then either ct;P or

g;P.

Remarks

a. Without statement 1, the remaining statements would

only refer to principal divisors.

b. Statements 4 and 5 contain the law of unique

factorization.

c. Statement 6 is often formulated as the "chain con­

dition."

d. Statement 7 is often referred to as the axiom of

prime divisors.

Some Resulting Theorems

1. Any divisor a can be split into prime factors P^ such

that

a = ^1 ^ 2 * • ■ ^h' 70

The prime factors are uniquely determined except for their

order.

Proof;

Suppose a is split into two factors P^ and Pg. Now suppose

P^ and ? 2 are split into two factors. Suppose that this process is maintained as long as there are still factors which are not prime.

But by Statement 6 in the section: Some More Explicit Statements, above, this process must come to an end. Therefore a can be split into a finite number of prime factors. Now if

a = a]_tt2 . . . is another prime factorization of a other than a = P2 P 2 • • • then by Statement 7 of the section: Some More Explicit Statements one of the factors P^, Pg, • . . P^ must be divisible by say P^;a^. Since

P^ is prime, P^ must be equal to and by cancelling the factor

Pj^ = « 2 we obtain

P^P^ . . . P^ — ^2^2 ....

We iterate this process until all factors are exhausted.

Therefore the factorization of a is unique.

2. A finite number of divisors « 2 » • • • “ h have a

greatest common divisor (gcd).

Proof:

Suppose we write e, e' “i = p ' V ^ . . . ®2 ®2 “ 2 = P P • • • 71

e._ e" = P V ^ . with a finite number of distinct prime factors P, P*, P", . . . and exponents e = 0. Set

e = minCe^, 6 2 » • . . e^),

e' = min(ej[, e^, . . . e^).

Then e ,e' a = p p . . . will be a common divisor of ag, . . . and any common divisor will be a divisor of a, written

a = (a^, 02, . . . a^). à 3. Any divisor a is the gcd of a finite set of numbers.

Proof :

By Statement 1 of the section: Some More Explicit State­ ments, we can choose a^ f 0 such that a^ is divisible by o. Set

“ 1 “ Then*aj^:a. Suppose o-j^ and o are not identical; then there exists an Integer ag divisible by a but not divisible by o^' Set

ctg = (a^, Og)

Continuing this process we obtain a chain of divisors o^îotg: . . . (:o) and a proper factorization

o, = 2 1 .2 ^ . . . ^ o o 2 3 of o-j^. Hence by Statement 6 of the section: Some More Explicit

Statements, this chain of divisors must stop after a finite number h of steps with 72

«h = “ • Therefore a is the gcd of a finite set of elements.

Suppose we look at some ideals in the ring of integers of

. Since

9 = 3-3= (2+/:5)(2-/3),

Kummer invented ideal numbers a and 6, for example, such that a|3 and a 12+/^; g I 3 and g 12-/^. Dedekind [33] considered the Ideals

A = (3,2+vC5) and

B = (3,2-y^).

By the definition of the product of two ideals,

AB = (9,

Every element of AB is divisible by 3 since all of the generators of

AB, 9, 6-3/^, 6 + 3 /^, are divisible by 3. In view of this,

AB à (3).

But observe that

3 = (6-3/3) + (6+3/3) -9eAB, that is,

(3) S AB.

Hence AB = (3). Therefore, we have factored (3) into the product of ideals. Now observe that

A^ = (9,6+3/3 ,6-31/3 , -1+4/3) - (2+/3) , because 9 = (2-/3) (2+/3), 6+3/3 = 3(2+/3), and -1+4/3 = (2+/3)^.

Also 2 + / 3 = 9-(6+3/3)-l+4/3sA^. Hence

A?- = (2+/3) 73 and similarly

b 2 = (2-/3).

Thus 3*3 = (2+/3)(2-/3) can be factored further yielding

AB-AB = A^b 2

and the factorization is the same. The question of whether or not A and B are prime ideals logically arises. Suppose we consider the ideal

A. In order to determine whether or not  is prime one needs to

determine whether or not a particular element is in A. For example, how can one determine if l+/3eA? Suppose we consider an arbitrary

element a+b/3. Observe the following; If

a+b = O(mod 3),

then a+b = 3n for some integer n. Now

a+b/3 = a+(3n-a)/3 = a+(3 (n-a)+2a) /3 = a(l+2/3)+3(n-a)/3eA,

because 1+2/3 and 3 are in A. Conversely, every element of A has the

form

(a+b/l5)3+(c+d/l5) (1+2/3) = (3a+c-10d) + (3b+2c+d)/3

and

(3a+c-10d) + (3b+2c+d) = 3a+3b+3c-9d = O(mod 3).

Thus a+b/3eA if and only if

a+b = 0 (mod 3).

Now since 1 f O(mod 3), then l^A, so A is proper. Now suppose

the product (a+b/3) (c+d/3) is in A. Then

ac-5bd+ad+bc = O(mod 3),

so

ac+bd+ad+bc = (a+b)(c+d) = O(mod 3) 74 because -5 = l(mod 3). But since is a field (see Appendix A), then

(a+b)(c+d) = O(mod 3)

implies that either a+b = O(mod 3) or c+d = O(mod 3) implies that

either a+b/-v5eA or c+d/3eA. Hence A is a prime ideal.

The above properties, though they exhibit a great deal of versatility, are only the beginning of many concepts associated with

this structure. (The reader is referred to Appendix A for other con­

cepts and examples of ideals which are conveniently stated in more modern terms.) Though neither Kummer nor Dedekind solved the Fermat

Problem, their work opened many other mathematical doors.

8 . Some Applications of Computer Techniques To Fermat's Last Theorem

The advent of the high-speed digital computer gave modern mathe­ maticians new hope and courage in their attempts to prove Fermat’s Last

Theorem. Though their work with computers was done long after Ideals

had been discovered, we shall digress from the development of Ideals to

cite some of the results. We shall see that many of the attempts to

prove this theorem, though fruitless, did contribute to our knowledge

of cyclotomie and algebraic number fields. We refer the reader to

Appendix A for definitions and explanations of basic concepts.

Before we discuss the applications of computers to the problem

of Fermat and the influence that Rummer's work had on these applica­

tions, we need to discuss Bernoulli numbers, a concept which Kummer

greatly utilized as we mentioned earlier in Section 3. Suppose we

consider the function 75

^ W(Z) = e"-l

But

e^-1 Z^ 1 + Z + — -+...+-1 implies that

Set 2 = 1 + B^Z + ^ Z" +

which implies that

2 B (1 + -|y + ^ + • •. . +) (1 + B^Z + ^ + . . , +) = 1.

If we multiply the two power series on the left, we must obtain a power series whose constant term is equal to 1 and whose remaining coefficients are all equal to 0, This yields infinitely many equations of the form

, B . B - 1 B. . 1 n , 1 n-1 . .11,1 _n. _in Ï T ÏÏT 2riK=ï)T + • • • T T + w T T -

After multiplying by (n-1), the binomial coefficients of the (n+l)-th power appear here on the left. They are

2B^+1 - 0

3Bg+3B^+l = 0

4B^+6Bg+4B^ +1=0 76

53^+105^+1082+53^ = 0

We obtain successively from these equations that

Bi = - I .

B 3 = 0 ,

" 30 ’

®6 = 3 *

DEFINITION 11-19. The numbers Bg, B^, . . . , defined above, are called Bernoulli numbers.

Remark

With the exception of B^, all B^ with odd index n have the value

0. For this reason, these numbers are often re-ordered so that

?•

X ®2 “ 30 ’ 1 ®3 “ 42 ’ 77

DEFINITION 11-20. Let p be a rational prime and let h be the class number of the cyclotomie field of pth roots of unity (see

Appendix A); then p is said to be a regular prime if p|h.

Remark

The small primes are all regular, (Thirty-seven is the first irregular prime.)

Kummer[116] discovered a simple test which he used to determine the regularity of a prime; in fact, he used this test as the definition of regular prime.

Rummer's Test for Regular,Primes : A prime p is regular if it does not divide the denominators of any of the first (2^) Bernoulli numbers of even index.

Kummer[116] in 1850 proved that Fermat's Last Theorem was valid for regular primes. H. S. Vandiver and some of his colleagues utilized Rummer's proof for regular primes in their many attempts to solve the Fermat Problem.

During the period (1928-1936) Vandiver[137] directed several collaborators at the University of Texas in examining the primes < 619.

They were successful in checking the validity of Fermat's Last

Theorem for the 36 irregular primes p such that p < 619. In 1953,

Vandiver decided to utilize the high-speed calculating machine of the

National Bureau of Standards due to the fact that the calculations had become too long and too laborious for a standard desk calculating machine. This machine (computer) known as SWAC and located at the

Institute of Numerical Analysis in Los Angeles was used to calculate 78 the irregular primes p < 2,000 via some results which we shall call

"Kummer's Criteria." These results are actually modifications of

Kummer's original results; these modifications were necessary in order to make them applicable to high-speed computation. The criteria are considered over a cyclotomie field K(a), where a is a primitive pth root of unity; ^i i 2 Ô k^ = a - a *

ki+1 Gi = ~ ’ where r is a primitive root of p and

(p-3) 2 -2in =1

(Note that the terras cyclotomie field, algebraic field, primitive root, etc. correspond with the definitions outlined in Appendix A.)

Rummer's Criteria:

1. If p is a given regular prime, then

jjP+yP+gP - 0 )

is impossible for rational integers x 0, y f 0 ,

z f 0.

2. If none of the units E^, with a = Og, . . . ag, is

congruent to the pth power of an integer in K(a) modulo

"q, where q" is a prime ideal divisor of (q) with q a 2 rational prime

integer; then

x^+yP+z^ = 0 79

is impossible in non-zero rational integers x, y, and z,

where again p is a regular prime.

In order to utilize the criteria stated above, the computer had to be programmed so that it could determine whether or not p was regular; if it was not, then it could determine all the s distinct aj^, ao, . . . a„ in the set 1, 2, 3, . . . • such that “ I

W % ------is each divisible by p. This had to be accomplished by examining the first Bernoulli numbers B^(mod p) via one of the many con­ gruences expressing Bernoulli numbers (mod p) as sums of like powers such as g g = = (-l)*(2 P-2 a_i)(3 p-2 a_2 P-2 a ^ , where

p>7, 2a

fg = (2P"^®-1) (3P"^®-2P“^ M ) is divisible by p which leaves the divisibility of B^ in doubt in such cases. However, if $ O(mod p) for all a, then p is regular.

D. H. and Emma Lehmer[137] programmed the computer so that it calculated Sg(modulo p) for all a = 1, 2, . . . in succession and punched out the values of p and a for which = O(mod p). They also instructed the computer to calculate the sum

p S = E S (mod p) , p = a=l ^ 2 and to test whether or not S = 0 (mod p) before proceeding to the next prime p. Now if S ^ O(mod p), SWAC was programmed to stop. 80

After the first run, the output, in the form of cards, was fed back into the computer. The computer was then programmed to test whether or not f^(modulo p) was divisible by p. If this factor were not divisible by p, then SWAC punched out another card with the values of p and a together with a code word indicating that the prime p was irregular with index a. If f^ = O(raod p), the Lehmers had coded the program so that SWAC then examined the congruence B S' = S S^^"^+ S S^^"^ = (-l)^fl^ ' (p>5, 2a

f’ = 6P"2a_5p-2a_2P-2a^l a was not divisible by p. If S^ was not divisible by p, then neither was and this particular outcome was disregarded due to the divisi­ bility of fa by p. If S^ = O(mod p), then p was irregular with index a and was recorded on an output card. But if the factor fg was divisible by p as well as f^, then it was necessary to resort to the longer formula of (y^) terms

Sg = E (p-2r)^^ = (-l)^'^2^^"\Bg(mod P^). r=l

Since it was determined that the necessary calculations could be 2 obtained modulo p , the computer was instructed to check that t* u 2 Sg 5 O(mod p) and punch a card every time S^ = O(mod p ).

The output of the above program revealed that there are 184 regular and 118 irregular primes less than 2,000. Below is a distribution table of these primes. Note that the interval up to 81

2,000 has been, divided into eight equal parts and 250k < p < 250(k+l) k = 0, 1, 2, • • , 7»

k= 0 1 2 3 4 5 6 7 Total

No. of irregular primes 9 19 20 16 11 14 11 18 118 No. of primes 52 42 37 36 36 35 33 31 302 Percentages, irregular primes 17 45 54 44 31 40 33 58 39

In the remaining part of the computation it was necessary to replace the quantity by a simpler quantity in K(a). In so doing, the Lehmers and Vandiver[137] were able to state and prove the follow­ ing lemma:

LEMMA II-l, Let t be any integer such that t^ t l(mod r), where r is a prime of the form r = kp+l

i . iP-2a^2P-2a^ . . . +

Then the unit Eg is congruent to the pth power of an integer in K(a) modulo a prime ideal dividing (q) if and only if k , -, Qg = l(mod q).

This lemma enabled them to state and prove the theorem below which is the main result of their work.

THEOREM 11-17. Let Qg be defined as in Lemma II-l and let

^a^' * • * ®ag be the only Bernoulli numbers, with indices less than , which are divisible by p. Then if

= l(mod p) 82

holds for i = 1, 2, . . . s, Fermat's Last Theorem holds for the

exponent p.

The theorem was then coded on SWAC which printed out the values

of p, 2 a, r, Qg and Q^ for all irregular p's < 2 ,0 0 0 , wherepis the odd

prime exponent in

xP+yP = %P

r = 1 +kp,

~ P ^p-l-2 a

and 2a is twice the index of the Bernoulli numbers defined in the

theorem. For example, if p = 547, the computer output, in tabular

form. gives us the Information that the Bai's of p consist of Bi3 5 and

Below is a partial table of SWAC output : ^243*

TABLE 1

Q k 2 a r 2 a P Qa Qa P r Qa a

37 2 2 149 146 81 257 164 1,543 1,371 1,258 59 44 709 137 645 263 1 0 0 1,579 1,154 1,385 67 58 269 73 180 271 84 1,627 951 367 1 0 1 6 8 607 514 47 283 2 0 1,699 1,388 383 103 24 619 273 389 293 156 587 8 8 113 131 2 2 263 91 128 307 8 8 1,229 266 151 149 130 1,193 420 178 311 292 1,867 523 360 157 62 1,571 1,293 1 0 2 347 280 2,083 1,561 1,711 157 1 1 0 1,571 170 1,261 547 270 5,471 1,797 3,692 233 84 467 276 55 547 486 5,471 4,814 1,591

As was stated earlier, in Section 1, each attempt to solve the

Fermat Problem generated some new mathematics in spite of the fact

that it did not solve the problem. The tabular print-out of SWAC con­

stituted new information about cyclotomie fields, especially for the study of units and ideals in a cyclotomie field for prime p < 2 ,0 0 0 . 83

In order to determine the p's, 2a's, r's, Qg's and Q^'s listed in the above table (Table 1), it is necessary to calculate Baj^, Bg^,

. . . Bg^ and E2 , . . . Eg. Hence the first Sg Bernoulli numbers associated with a cyclotomie field for p < 2 , 0 0 0 are determined.

Similarly, the number of units, the fundamental system of units, and independent sets of units associated with cyclotomie fields for p < 2 , 0 0 0 may be determined when the E^, E2 , . . . Eg are calculated.

Other information associated with cyclotomie fields for p < 2,000, which may be obtained, includes the determination of the class number h of such fields, the pth roots of unity, and of course, the regular and irregular primes.

In 1954, Vandiver[181] was able to extend the results that he and the Lehmers had obtained on the Fermat Problem. With the help of

John Selfridge, who carried out the calculations on the SWAC, Vandiver was able to prove the impossibility of a non-trivial

xP+yP = zP for regular primes p such that 2,000

Vandiver utilized the methods and procedures which were outlined in the earlier paper[137]. The computer print-out, again in tabular form, re­ vealed that there are 39 regular primes and 26 irregular primes p such that 2,000

Vandiver was an avid student of Fermat's Last Theorem. He had written several papers on this problem prior to his investigations with

SWAC. Realizing the potential of this computer, he was very eager to utilize it on the'Fermat Problem. In 1955, he again extended the proof of the celebrated theorem for regular prime exponents.

Vandiver[186] collaborated with C. A. Nicol and John Selfridge, who made the calculations on SWAC under the general supervision of

C. B. Tompkins. They basically utilized the procedures stated earlier to extend the prime exponent p to the range 2

Irregular and regular primes to 216 and 334 respectively. The total

216 represents a percentage of 39.3 irregular primes. As stated before, these findings yielded new information for the theory of cyclotomie fields in terms of Bernoulli numbers, number of units, fundamental systems of units, pth roots of unity, regular primes, and irregular primes.

Euler, one of the world's greatest mathematicians, attempted to solve the Fermat Problem as was briefly mentioned in Section 1.

Though he, as well as all others, failed to prove the theorem, Euler

[49] made a more general conjecture that includes our famous problem as a special case. Euler suggested that no nth power greater than 2 can be the integral sum of fewer than n nth powers. Of course, this 85 problem or conjecture has been verified for n = 3. It is Fermat's Last

Theorem with exponent 3. To date, it is not known whether

x^+y^+2^ = a^ has a solution. According to Euler, there should be four fourth powers, i.e.,

A^+B^+C^+D^ = e '^ should have solutions. It was not unitl 1911 that a solution to the above equation was found by the mathematician Norrie[l65].

In 1958 with the aid of a computer, John Leech[133] was able to generate several such solutions by using Norrie's work and the fact that any fourth power is congruent to 0 or 1 modulo 16. Below is a table of solutions which includes Norrie's solution as the first line or smallest solution:

TABLE 2

A B C D E

30 1 2 0 272 315 353 240 340 430 599 651 435 710 1,384 2,420 2,487 1,130 1,190 1,432 2,365 2,501 850 1 , 0 1 0 1,546 2,745 2,829 2,270 2,345 2,460 3,152 3,723 350 1,652 3,230 3,395 3,973 205 1,060 2,650 4,094 4,267

In 1966, however, L, J. Lander^and T. R. Parkin[128] found a counterexample to Euler's conjecture. By use of the CDC 6600 computer, these mathematicians found that

27^+84^+110^+135^ = 144^. 86

This is the smallest instance in which four fifth powers sum to a

fifth power, thus contradicting Euler's conjecture that at least five

fifth powers are required to sum to a fifth power.

In Section 1, a proof for the validity of a solution for

x^+y2 - 22 was given. This, of course, is the Fermat Problem for n = 2. The versatility of the high-speed computing machines motivated mathematicians

to consider some of the ramifications of this special case. In particular, M. Lai and W. J. Blundon[124] became Interested in systems

of such diophantine equations due to their geometric significance.

The equation

X V =

is the well-known Pythagorean Theorem of plane geometry. The system of equations

x^+yZ =

y 2+ 2 2 = 1^2

z^+x^ = n^ has solutions which represent rectangular parallelepipeds whose edges and face diagonals are all rational integers. It is known that if

(x,y) = (y,z) = (z,x) = 1 ,

the system of three equations above has no solution; otherwise there

are infinitely many. The proof of

x V - z2 2 2 yielded the primitive solutions of the form x = 2a$, y = a - 6 ,

z = «2+^2 with (a,3) = 1 and a*3 = 2 (mod 4); 87

Lai and Blundon. used algorithms developed by Rignaux[177l to generate the solutions to their system of diophantine equations. Namely,

X = 2mnpq; y = mn(p2-q2); z = pqCra^-n^).

One observes that

x2+y2 = m2n2(p2+q2)2

y2+z2 = m 2n2(p2-q2)2(jij2_g2j2

z2 +x2 = p2 q2 (m2 +n2 ), and

that each right hand member is the square of an integer and is there­

fore an integer. One also can observe a similarity to the solutions of the Fermat Problem for n = 2. Below are a few of the values for

X, y, and z taken from the table given in the paper[124] by Lai and

Blundon.

X Y Z X Y Z 44 117 240 195 748 6,336 85 132 720 240 252 275 140 480 693 429 880 2,340 160 231 792 495 4,888 8,160 187 1 , 0 2 0 1,584 528 5,796 6,325

We conclude this rather lengthy section on some of the computer

techniques applied to Fermat’s Last Theorem by briefly mentioning the work done by A. S. Fraenkel. In 1969, Fraenkel[52] made a study of

the equation

T^(6 ) = x(x+6)(x+26) . . . (x+(m-l)6 ), where m and 6 are arbitrary but fixed positive integers, m >1. He was

particularly interested in the special case of this equation T“(S)+TyCô) = tJ(6)

for the cases ra = 2 , 3. Fraenkel observed that for 6 = 0 the solu­

tions for the above equation are the so-called "Pythagorean Triplets." 88

He showed how the solutions of the equation for various values of m and 6 ge^ erate triangular and tetrahedral numbers via high-speed compu­ tation. Again, this is a ramification of the Fermat Problem.

One observes, on the basis of the proceeding examples, that the computer techniques generated a great deal of mathematics, even in modern times, as a result of their application to this problem. Now that this point has been made, we shall return to the development of ideals.

We digressed from the main "train of thought," in Section 8 in order to demonstrate the power of Fermat's Last Theorem and the effects that it had on mathematics even in modern times. As was in­ dicated earlier, each attempt to solve the problem, though unsuccessful, generated a great deal of mathematics. If we return to the development of the ideal theory which evolved from algebraic number fields, directly.our next link would normally be the work of Emmy Noether and her students with one exception.

Artin, as we shall see, was not a student of Noether though he was somewhat influenced by her and certainly her works. However, Artin was also linked to Wedderburn. In fact, Artin more or less fused the works of Wedderburn and Noether which resulted in what is now known as Artinian rings. (It will be interesting to observe later than van der Waerden was somewhat similarly linked to Noether and Artin.) More particularly, Artin, as well as some other mathematicians discussed in

Chapter V, was greatly inspired by Wedderburn's work on hypercomplex numbers. For this reason, it is necessary to inject a discussion of

Wedderburn and his work on hypercomplex numbers at this point. 89

9. Wedderburn and His Hypercomplex Number Systems

Debus[32] gives a brief account of the life of this famous

algebraist. Joseph Henry Maclagan Wedderburn (1882-1948) was born in

Forfar, Scotland on February 26, 1882. He received his M.A. and

D. Sci. in mathematics in 1903 and 1908 respectively from Edinburgh

University. He later studied mathematics at the universities of

Leipzig, Berlin, and Chicago. After a brief period as an assistant in mathematics at the University of Edinburgh, he became associated with

Princeton University in the United States; he.remained until his

retirement in 1945, when he was named professor emeritus. Though

Wedderburn’s major mathematical contributions were in the areas of matrices and algebras, the influence which he had on the development

of "modern algebra" was almost phenomenal.

Artin[5] in 1950 attempted to give an account of the influence

that Wedderburn had on the development of "modern algebra." We shall

summarize some of Artin's analysis of Wedderburn*s influence in the

paragraphs which follow.

In 1908 Wedderburn[214] wrote a paper on hypercomplex numbers which was one of the major "turning points" in the development of ideal

theory and modern algebra in general. In order to appreciate this paper fully, we have to consider briefly the works of earlier mathematicians on this topic.

There appeared to be a distinct difference in the attitudes of

American and European authors on the subject of hypercomplex numbers.

The American papers seemed to have emphasized the abstract point of view while the European papers seemed to have emphasized a system of 90 hypercomplex numbers as an extension o£ either the real field or the complex field. In view of this difference, the Americans achieved an abstract formulation of the problem which developed a suitable termin­ ology and discovered the foundations of modern methods. On the "other side of the coin," the Europeans obtained very advanced results in the classification of their special cases with methods that were not well adapted to generalizations.

The leaders on the American side were Peirce[170], Hawkes[77]^and

Dickson[37]. Peirce was the leader of the Americans with his paper of

1870 which was published by the American Journal of Mathematics after his death. Hawkes was responsible for putting Peirce's results in a more readable form. Dickson stated the first correct definition of an associative algebra over an arbitrary field.

At the same time in Europe, the leaders of the movement there were Cartan[19], Frobenlus[53]^and Molien[156]. Each of these mathe­ maticians arrived at many of the results of the modern theory of hypercomplex number systems or algebras. Cartan derived the structure of simple algebras without recognizing the possibility of stating the result in the very simple form which Wedderburn discovered. All of the above authors were very proficient with manipulations in the complex field and did not hesitate to use them. Fortunately Wedderburn capitalized on both trends of thought,

Wedderburn[214] was successful in synthesizing the American and the European lines of investigation. He extended the proofs of all the structure theorems found by the European mathematicians for the special cases of the real and complex fields to the case of an arbitrary 91 field. By utilizing the "calculus of complexes" combined with

Peirce's "decomposition relative to an idempotent," he was able to prove his theorems within a given field and in a simpler way. He was the first to find the real significance and meaning of the structure of a simple algebra. Wedderburn's work on algebras is the "link" between

Dedekind's divisors and Emmy Noether's ideals in the decade of the

1920's. This will be discussed in Chapter V. An explanation of algebras or hypercomplex numbers, as they were sometimes called, is certainly in order.

At this point the questions, "What is an algebra?" and "How are algebras related to ideal theory?" are also in order. Albert[2] gives an excellent and modern discussion of algebras and the major theorems that Wedderburn proved about them. We shall draw upon his analysis in our discussion below.

DEFINITION 11-21. Let F be a field. A linear set U of order n over F is the set of all sequences (a^^, a2 , ag, . . . a^) with a^ in F such that

f ^n^ bg, bg, . . . ^bg) — (a^^+bj^, U2+b2, . . . j

^n+'^rA-

Sg, . . . a^) = (a^, a^, . . . a^)k = (ka^, ka^, . . . ka^) for all a^, b^, ksF.

DEFINITION 11-22. Let F be a field with unity 1, R be a ring and let the scalar products

ar = ra be in R for every a of F and r of R. Then we call R an algebra of order n over F if for every a and b of F; r, seR: 92

1. Ir = r, a(br) = (ab)r, (ar)(bs) = Cab)(rs);

2 . (a+b)r = ar+br; a(r+s) = ar+as,

3. There exist elements r^, rg, . . . in R such that every

element r of R is uniquely expressible in the form

r = airi+a2 r 2 + . . . +a„r„

for a^ in F.

Remarks

1. It is obvious from the definition that an algebra is a linear set.

2. The algebra defined above is also called a finite-dimensional algebra and throughout this section algebra will be used to mean finite-dimensional algebra.

DEFINITION 11-23. A subalgebra S over F is any linear subset which is an algebra with respect to the operations defining Ry that is, a linear subset S of R is a subalgebra of R if and only if the product of any two elements of S is in S.

Wedderburn[214] stated a theorem which is most significant to his work on algebras. In fact, this theorem will be utilized as a tool theorem later in this section. (We invite the reader to refer to

Appendix A for the basic material on algebras and related topics.) The theorem is

THEOREM 11-18. Let the unit of a total matrix subalgebra M of an algebra R coincide with that of R. Then

R = M % S,

S = R®, 93

A proof of this theorem is given by both Wedderburn[2l4] and

Albert[2].

Since an algebra is defined to be a ring with several additional properties)then a logical question would be "What is the ideal struc­ ture of an algebra?" Certainly, Artin must have entertained this question. For this reason we explore it now.

DEFINITION 11-24. A linear subset S of an algebra R over a field F is called a left ideal of R if RS S S.

Remarks

1. Since SS - RS - S, then every non-zero left ideal of R is a subalgebra of R.

2. A right ideal of R can be defined analogously.

3. A two-sided ideal or simply an ideal of R is a linear subset of R which is both a left and a right ideal of R.

4. R, R^, R^, . . . are all ideals of R.

DEFINITION 11-25. An algebra R is said to be nilpotent if RP = 0 for some exponent p and the least such p is called the index of R.

THEOREM 11-19. All subalgebras of a nilpotent algebra are nil- potent. Moreover R has a non-zero proper ideal if and only if R is not a zero algebra of order one.

Proof;

Suppose Rq is a subalgebra of a nilpotent algebra R of index a.

Then RqCT R and R“ = 0 implies Rq = 0. Therefore, Rq is nilpotent.

Further, suppose R is an algebra of order one; then R has no non-zero proper subalgebra. Also suppose that R is a nilpotent algebra of order n>l. If R is a zero algebra, then R = (r^, tg, . . . r^) over F and 94 r^Zj = 0 for i,j = 1, 2, . . . n. Hence the algebra (r^, r2 »

‘ 2 r^_^) is a non-zero proper ideal of R. Otherwise R f 0 is a non-zero proper ideal of R,

DEFINITION 11-26. A left (right or two-sided) ideal S of an algebra R which is either zero or a nilpotent algebra is called a nilpotent left (right or two-sided) ideal.

Remark:

All ideals of a nilpotent algebra are nilpotent ideals.

Wedderburn[214] proved the following lemmas;

LEMMA II-2. The sum of two nilpotent left ideals S and T of R is a nilpotent left ideal of R.

LEMMA II-3. If L is a nilpotent left ideal of R, then the sum of

L and LR is a nilpotent ideal of R.

LEMMA II-4. Every nilpotent left, right, or two-sided ideal of an algebra R is contained in a unique maximal nilpotent ideal of R.

In view of Lemma II-4, we have

DEFINITION 11-27. The radical q of an algebra is its unique maximal nilpotent ideal.

LEMMA II-5. Every non-nilpotent algebra R contains an idempotent element.

DEFINITION 11-28. A left ideal S f 0 of R is called left simple if there exists no left ideal T of R such that 0«= Tc S.

LEMMA II-6 . A left simple left ideal L of an algebra R is either a zero algebra or L = Re for some idempotent e of L. 95

Peirce[170] observed the following in his investigations of algebras via an indempotent e.

Define Le to be the set of all elements x of R such that xe = 0.

Now, if ye = 0, then a(ye) = 0 and (ax+By)e = 0 for a in R, a and 6 in

F. Hence the set Le is a left ideal of R. Now observe that

a - ae+(a-ae) for any a of R. Now since e is an idempotent e^ = e and hence (a-ae)e = ae-ae^ = ae-ae =* 0 implies a-ae is in Le. Further ae is in Re. Now if b is in both Le and Re, then Re = 0 and be = b imply b = 0. Therefore,

R = Re+Le.

DEFINITION 11-29. Re and Le are said to be supplementary if their sum is R.

Remarks

1. a = ae+(a-ae) is called the left-sided Peirce decomposition of a.

2. R = Re+Le is called the left-sided Peirce decomposition of R relative to e.

In view of the contributions of Peirce, we have that

LEIQ'EA II-7. Let e be an idempotent of an algebra R. Then

R = eRe+eLe+Re+Ce, where Re is the set of all x in R such that ex = 0, and Ce is the intersection of Re and Le.

Remarks

1. eRe, eLe, ReC and Ce are all algebras.

2. eLe is the set of all elements beR such that eb = b, be = 0.

3. Re is the set of all elements c of R such that ce = c, ec = 0. 96

4. eRe Is the set of all d of R such that ed = de = d = ede.

5. C Is a left Ideal of R.

LEMMA II-8 . Suppose Q is the radical of R and e is an idempotent of R. Then the radical of eRe is the intersection eQe of Q and eRe; the radical of Ce is the intersection of Q and Ce. (See Appendix A for definition of radical.)

DEFINITION 11-30. An idempotent e of an algebra R is called a principal idempotent of R if R has no idempotent r orthogonal to e.

Remark

In the Peirce decomposition of R relative to e, the set Ce con­ sists of all elements orthogonal to e.

LEMMA II-9. An idempotent e is principalin R if and only if Ce is zero or a nilpotent algebra.

LEMMA 11-10. If u is a non-principal idempotent of R, there exists a principal idempotent e = u+v such that eu = ue = u and v. is an idempotent orthogonal to u.

LEMMA 11-11. Every non-nilpotent algebra contains a principal idempotent.

LEMMA 11-12. If e is a principal idempotent of R, the set Le is contained in the radical of R.

DEFINITION 11-31. An idempotent e of an algebra R is called primitive in R if e is not the sum u+v of orthogonal idempotents u and

V.

LEMMA 11-13. An idempotent e is primitive in R if and only if R contains no idempotent r e such that er = re = r; that is, e is the only idempotent of eRe. 97

LEMMA 11-14. Every non-primitive idempotent e of an algebra R is the sum of a finite number of pairwise orthogonal primitive idempotents e^ of R such that e^e = ee^ = e^.

We leave the lemmas (theorems) pertinent to idempotents long enough to have a look at the decomposition of algebras.

DEFINITION 11-31. An algebra is defined to be the direct sum of algebras R^ and we write

R = Rj (±) R2 © . . . ® R^ if the R^'s are subalgebras of R such that

R = Rj_+R2 + . . . +R^ and R^R^ = 0 for i f j.

Remark

RRj^ S R^, R^R S R^ and the R^ are ideals of R.

DEFINITION 11-32. An algebra R is called an Irreducible algebra if it is not expressible as the direct sum Rj^ ® R 2 © • . . © of t ;> 1 subalgebras R^. Otherwise R is called reducible and the R^ are called components of R.

Remark

If R has a unit element, then so does every component R^ of R.

Continuing our list of lemmas, we have;

LEMMA 11-15. Let an algebra R with a unit element be expressed as a direct sum

R = Rj^<©R2 © . . . ©R ^ .

Then a linear subset S of R is a right, left or two-sided ideal of R if and only if 98

S = Si © S2 © . . • © Sj. j where the is correspondingly a right, left or two-sided ideal of R^.

LEMMA 11-16, Every reducible algebra with a unit element is uniquely expressible as a direct sum of irreducible components apart from the order of its components.

LEMMA 11-17. Let R be an algebra with unit element e such that

R is expressed by the direct sum

R = Sj^©S 2 0 . . . and let Q be its radical. Then

Q = Ql © Q2 © • • • © Qt » where = is the radical of S^.

LEMMA 11-18. Let an ideal S of an algebra R have a unit element e. Then

R = S ©C e .

Moreover, in any expression

R = S © C , we have that C = Ce.

DEFINITION 11-33. Let R be an algebra and S be a linear subset TJ of R. Then the quotient group — exists. In terms of additive notation

can be indicated by R-S and is called the difference group. In particular, if S is an ideal of order m over F and R is of order n over

F and we define coset multiplication by

[ail[8 2 ] = [3 1 8 2 ] in the difference group R-S, then R-S is an algebra of order t « n-m over F called the difference algebra of R modulo S. 99

LEMMA 11-19. Let R = S+T where S is an ideal of R and T is a

subalgebra of R. Then R-S is an algebra isomorphic to T.

LEMMA 11-20. Let S be an ideal of an algebra R, Eg = R-S. Then

there is a 1-1 correspondence between the subalgebras Tq of Rg and the

subalgebras T - S of R such that

T <-> Tq = T-S.

Moreover, Tq is an ideal of Rq if and only if the corresponding T is

an ideal of R, and in this case we have the equivalence over F,

R-T= Rq -Tq .

DEFINITION 11-34. An element y 0 of an algebra R is called properly nilpotent if both ay and ya are zero or nilpotent for every a of R.

LEMMA 11-21. An element y f 0 of an algebra R is properly nil- potent if and only if for every a in R one of the products ay, ya

is zero or nilpotent.

LEMMA 11-22. The set Rq consisting of zero and all properly nil- potent elements of an algebra R is its radical.

LEMMA 11-23, Every subalgebra S of a division algebra D (see

Appendix A) is a division algebra, whose unit element coincides with

that of D and whose order divides that of D. 2 LEMMA 11-24. Let an algebra R contain m elements e^j, not all

zero and satisfying

" Gik, eijehk = 0 (j # h, i, j, h, k = 1, 2, . . . m);

then the linear set Mn spanned by the e^j is a total matrix subalgebra 2 of order m of R with unit element

e = 6 1 1 + 6 2 2 + * • • +emm' 100

The twenty-four lemmas (theorems) exhibit the various properties of algebras. As has been our convention in the past, these lemmas are preliminaries to fundamental theorems. The fundamental theorems here are the Wedderburn structure theorems. It was these theorems that

Artin[6] considered in his study of rings satisfying the maximal and minimal conditions which we shall investigate in Chapter IV.

THEOREM 11-20. Every serai-simple algebra (see Appendix A) has a unit element.

Proof :

A serai-simple algebra by definition is not nilpotent. But by

Lemma 11-11, every non-nilpotent algebra contains a principal idem- potent e. By Lemma 11-12, Le is contained in the radical of R, Q = (0).

But

Le = {xcRjxe = 0}ct(0) and since e is principal implies that Le =Q"(0). But Le ,= (0) implies

THEOREM 11-21. Let S be a nilpotent ideal of R^S. Then R-S is semi-simple if and only if S is the radical of R.

Proof : = >

Let S be the radical of R-S, and let S^ be the corresponding sub­ algebra of R such that

? = S^-S.

By Lemma 11-20, is an ideal of R. Now = 0 so that - S, O (S|) = 0, Sj^ is a nilpotent ideal of R. If S is the radical of R, then

S^ - S, S = 0, R-S is semi-simple. 101

Proof ; < =

If R-S is serai-simple and S^ is the radical of R, we .have that

S^-S is a nilpotent ideal of R-S; thus S^-S = 0 and S^ = S.

DEFINITION 11-35. An algebra R is called simple if the only proper ideal of R is the zero ideal and if R is not a zero algebra of prime order.

In view of this definition, we have the following short theorems:

THEOREM 11-22. A simple algebra is semi-simple.

Corollary: A simple algebra has a unit element.

DEFINITION 11-36. A proper ideal S of R is called maximal if no proper ideal T of R contains S.

In view of the above, we have a theorem which is true for algebraic structures such as groups, rings, modules and vector spaces.

THEOREM 11-23. Let S be an ideal of R. Then S is maximal if and only if R-S is either simple or a zero algebra.

Applying Lemma II-8 with Q = (0), we have the theorem;

THEOREM 11-24. If e is an Idempotent of a semi-simple algebra R, then the algebra eRe is semi-simple.

Continuing our discussion we have

THEOREM 11-25. If e is an idempotent of a simple algebra R, the algebra eRe is simple.

It is a fact that a division algebra D has only one idempotent, its unit element. This is observed by designating e and u as the unit element and idempotent of D respectively. Then -1 2 uu = e, = u 102 implies that

u~^(u^) = eu, which implies that

eu = u, which implies that

u“^u = e.

Hence u = e. In view of this fact, we have the significant structure theorem which is often used as a tool theorem.

THEOREM 11-26. Let e be an idempotent of a simple algebra R.

Then eRe is a division algebra if and only if e is primitive in R.

Proof; — >

Suppose eRe is a division algebra; then e is the only idempotent of eRe. But by Lemma 11-13, e is primitive in R.

Proof: < =

Suppose e is a primitive idempotent in R so that

D = eRe is a simple algebra with e as unit element by Theorem 11-25. Let a OeD; then Da is a non-zero left ideal of D. But since a simple algebra is semi-simple, then D has a zero radical; thus Da is not nilpotent. Hence Da contains an idempotent which must be

e = ba

for some beD. Thus the non-zero elements of D now form a multiplica­

tive group and D is a division algebra; that is, eRe is a division algebra. 103

THEOREM 11-27. A semi-simple algebra R is irreducible if and only if it is simple.

Proof ; = >

Suppose R is irreducible but not simple so that R has a non-zero proper ideal S. Since R is semi-simple, it has a unit element and hence

RS — SR — S.

Also if Q is the radical of S, then the set RQR - S and RQR is an ideal of R. But SQS - Q and is nilpotent, and also SQS is an ideal of R, therefore SQS =0 . In view of this, we have

(RQR)3 5 (RQR)*Q-(RQR) S SQS = 0,

RQR is a nilpotent ideal of R and RQR = 0. But since R has a unit element

RQR- Q, Q = 0, and S is semi-simple. S then has a unit element. Then by Lemma 11-18,

R = S ©Ce,

R.3S, Ce ^ 0. But this is a contradiction because Ce = 0 since R is irreducible. Hence R is simple.

Proof : < =

Suppose R is a serai-simple algebra such that R is also simple.

Suppose also that R is reducible. Since R is reducible, then R can be expressed by the S^ components

R = Sj^ © S2 © . . . © S ^ .

Now R is semi-simple implies that R has a unit element. But if R has a unit element, then each S^^ has a unit element. This implies that R 104 has a non-zero proper ideal. This is a contradiction since R is simple. Hence R is irreducible.

Semi-simple algebras may be characterized by the following structure theorem.

THEOREM 11-28. An algebra R is semi-simple if and only if R is either simple or is expressible as a direct sum of simple components.

These components are unique except for their order.

Proof : ==*=>

Suppose R is semi-simple. By Lemma 11-16, R is either irreducible or is the direct sum

R = R^ @ Rg # . . . $ Rg. for the R^ irreducible ideals of R with respective unit elements e^ such that e2 +eg+ . . . e% is the unit element of R. But if R is ir­ reducible, then it is simple. Otherwise t>l and R is not simple. But

^i “ Ci**i and is semi-simple by Theorem 11-24. But by Theorem 11-27, each R^ is simple. Using Lemma 11-16 again, we see that the Rj^ are unique.

Hence R is either simple or expressible as a direct sum of simple components.

Proof ; <==

Suppose R is a direct sum of simple algebras so that the sum of the unit elements of its components is the unit element of R, Each component has a zero radical and by Lemma 11-17, R has a zero radical.

Hence R is semi-simple.

The next and final theorem and its revealing corollary are among the most profound results concerning algebras. Wedderburn, upon 105 discovering this theorem, became the first to find the real significance and meaning of the structure of a simple algebra. This theorem is

THEOREM 11-29 (Wedderburn). Every simple algebra R is expressible as

R = M X D, where M is a total matrix algebra and D is a division algebra. Con­ versely, every such direct product is simple. Moreover, if the unit element of R is expressed as a sum e^+eg+ . . . +e^ of pairwise orthogonal primitive idempotents e^ of R, then M may be (e^^; i, j = 1,

2, . . . m) with e^^ = e^ for i = 1, 2, . . . m.

Proof;

By Lemma 11-14, R has a unit element 1 = e^+Sgf . . . +e^, where the e^ are pairwise orthogonal primitive idempotents of R. The algebras e^Re^ are division algebras by Theorem 11-26. Now sinceRe^R is an ideal of R containing e^, then we have that

Re^R 4 0, Re^R = R for i = 1, 2, , . . m.

Define

^ij “

Then eye^ = e^, e^e^^ = 0 for j 7* h and we have that;

(1) RijRjk “ R^R^^ - 0 (J / h; 1. j. h. k -

1, 2, . . . m) implies that the product R^jR^j = 0, or

R^jR^j = R^j. The R^^ are subalgebras of R with the

properties

(2) e^a^j = a^jSj = a^^, e^a^^ = ®ij®k = 0(h ^ i* k ^ j; i, j,

h, k = 1, . . . m), for every a^^ of R^j, If j > 0 we have 106

that ^ll^^ij^ji ^ some of R^^. Then

*l»ji - »j f 0 in for some of R^^, Since is a division algebra,

then there exists an element bj in R^^ such that

bj*j = "1- By (1), the element b^a^^ = e^^ is in R^^ and since e^e^ = e^, we have proved the existence of the elements

in R^j » in R^^ such that

(3) ~ >1 (j = 1, 2, . . . m).

Define

(4) e^j = ®il®ij (i, j = 1, 2, . . , m)

and consequently obtain

:lj=jk ' 'll'tj=3i=ik ■ ^ l l V l k ° ®lk (i. j. k = 1. . . . m). Also e^j ee^Re_^ and by (2) we have that

®ij®hk ° (j k; i, j, h, k = 1, 2, . . . m).

Since e^^ = f 0 we can apply Lemma 11-24 to see that

H = (e^j, i, j = 1, 2, . . . m) 2 is a total matrix subalgebra of order m over F the field for R. Then by (5)

is an idempotent of e^Re^ and since e^ is primitive, it is the only

idempotent of e^Re^,,

®il ■ 107 is the unit element of both R and M, But by Wedderburn's tool theorem

(Theorem 11-26) R = M x D, where D = R^. R is the supplementary sum of the algebras e^^D and M x D is a direct product; the order of

is the same as that of D, The correspondence

Sj^d -M- d for d in D is thus a 1-1 correspondence which trivially defines an equivalence of the e^^D and D since e^ is an idempotent such that

e^d = de^ for every d of D. But, as we have already stated e^Re^ is a division algebra. This implies that D is a division algebra.

Proof: < =

Conversely, let R = M x D and S be a non-zero ideal of R. Then S contains a non-zero quantity

m

with elements d^j in D and not all zero. Hence if d^^ f 0, it has an inverse in R and since S is an ideal,

-1 ”

is in S for every a of R. Hence we have proved that S 2 R, S = R and that R is simple.

Corollary. A commutative semi-simple algebra is a direct sum of fields.

Artin[5] points out the fact that the first period following this significant discovery of Wedderburn was spent in polishing up his 108 proofs. By this time, the Europeans had discovered the advantage of the abstract point of view.which had been so emphasized very early by the

Americans of this period. We shall see in Chapter IV that Artin studied rings satisfying the minimal and maximal conditions in the same manner that Wedderburn studied his algebras. However, we shall digress from this pattern in the development of ideals to observe another "birth" of ideals, which was simultaneously occurring during the time of Dedekind’s

"birth" of ideals.

The m o d e m theory of ideals has two beginnings. The theory cited thus far grew out of the theory of algebraic numbers related to Fermat’s

Last Theorem. This theory is often referred to as multiplicative or classical ideal theory by modern algebraists. The other origin is due to the study of certain polynomial rings associated with algebraic geometry. This is sometimes called additive ideal theory. The next chapter concerns itself with this aspect of the foundation of ideal theory. CHAPTER III

IDEALS AND THEIR FOUNDATIONS IN

ALGEBRAIC GEOMETRY

1. Kronecker and His Theory of Forms

We shall appeal to Gillisple[60] as a reference source for

Kronecker's biography. Leopold Kronecker (1823-1891) was a student of

Kumraer. Kronecker was born into a rather wealthy family; his father, a successful businessman, provided private tutoring for him until he entered the Liegnitz Gymnasium. It was there at the gymnasium that he first encountered Kummer as his mathematics teacher. Kummer, who quickly recognized his exceptional ability, encouraged him to initiate independent research, gave him his basic mathematical foundations, and remained a life-long friend. As one would expect, Kronecker’s mathe­ matical works were greatly influenced by Kummer.

During the winter of 1862 Kronecker, at Kummer’s suggestion, made use of his newly acquired right as a member of the Berlin Academy.

This statutory right allowed him to deliver a series of lectures at the University of Berlin. He lectured primarily on the theory of algebraic equations, the theory of numbers, the theory of determinants,' and the theory of multiple and simple integrals. He attempted to simplify and refine the existing theories in the above topics and attempted to present them from new perspectives. His teaching and his research activities, however, were closely related. His research

109 110 activities were similar to those of his close friend Weierstrass in that they were most concerned with those mathematical ideas which were currently in the process of development. However, he differed from

Weierstrass in that he did not attract great numbers of students due to the lack of organization of his thoughts.

Until the middle of the 1870*s, Kronecker and Weierstrass had maintained a very close relationship. Though they continued to work together for a while after this period, their relationship began to disintegrate. The growing estrangement between the two men was actually accelerated by the interference of their mutual friends. Gillispie[60] points out the fact that Kronecker*s objection to Weierstrass* methods is manifested in well-known dictum that "God Himself made the whole numbers— everything else is the work of men.” In line with this,

Kronecker believed that all arithmetic could be based on the set of whole numbers. Weierstrass contended that the irrationals were needed.

Kronecker also believed that all mathematical disciplines except geometry and mechanics were arithmetical, where the terra arithmetical includes algebra and analysis.

During the early part of the 1880*s, Kronecker became interested again in some of the more algebraic theories of Kummer. Kronecker[105] was desirous of putting the divisors of algebraic integers in a simple and natural form so that the properties of Kummer*s ideal numbers could take on a more concrete mathematical mold. We shall see that he did this by a methodical use of indeterminate coefficients in such a way that the resulting set of algebraic quantities satisfies the funda­ mental laws of arithmetic, in particular, the laws of division. Ill

In a sense one can say that Dedeklnd and Kronecker were rivals striving for the common goal of finding a general arithmetic for algebraic number fields. (This rivalry was in no way comparable to the one between Kronecker and Weierstrass because they were aware of their competition and actually became enemies.) Weyl[215] analyzed and compared the methods of Kronecker and Dedekind as follows: Recall that in the ring of integers of the field there was no unique factorization of a divisor into prime divisors. Dedekind’s remedy, we recall, was to extend the notion of a divisor d so that it (the divi­ sor or ideal) characterized a set S of rational integers, so that if dja, that is, if aeS, then d|Xa for any rational integer X, that is,

XaeS for any rational integer Xj; and if dja and d|b; that is, if a, beS, then d|a+b which says that aibeS. In other words, the integers divisible by d form a divisor (ideal).

Kronecker's theory was characterized by the systematic use of indeterminates. He Investigated certain polynomials called forms of

indeterminates x^, X2 . . ., whose coefficients are algebraic integers of a ground field k. He observed their behavior in a finite extension field L over k; that is, he observed the properties of elements of k(x2 , X2 > • . . ) which remained invariant in L(xi, X2 » . . .) and vice versa. Of course, he was particularly interested in the divisibility properties of the rings of Integers in these fields. Kronecker[105] utilized the concept of "content" in his investigation. This most crucial concept will be formally defined in Section 2 of this chapter.

We shall also see in Section 2 that Kronecker defined his integers or integral elements in terms of their content and that these integers 112 have a ring structure which is comparable with the integers in

Dedekind’s theory. Kronecker centered his theory of divisibility around the polynomial functions which were called forms. We shall see that the content of a form plays the same role that a divisor (ideal) does. We shall pause here to explore some of the fundamental concepts of forms following Hancock[74].

2. Some Fundamental Concepts of Forms

DEFINITION III-l. A function (linear combination) f(xj^, Xg*

. . • jX^j) of indeterminates x^, X2 , . . *jX^ whose coefficients aj^, a2 ,

. . . j,a^ are the algebraic integers of a fixed field k is called a form of the field.

Remarks

1. The indeterminates x^, X2 , . . .jX|^ are more or less place­ holders and are of little mathematical significance.

2. Kronecker focused his attention on the set of coefficients

32» * • 'f&r'

DEFINITION III-2. Let f be a form whose coefficients are a2 ,

. . .jaj.. If each a^ is replaced by its conjugate then each re­ sulting form f(^) is called a conjugate form; that is, each of the f(l) f(2 ) f(3). , . is called a conjugate form, where n is the

degree of the particular ground field k over 0 .

Remarks

1. Kronecker[105] showed that

f.f(l).f(2 ) _ ^ , f(*"l) = aX,

where X is an integral function (see Appendix C) of x^, X 2 , . . . with rational integral coefficients and where the rational integer a has 113 been so chosen that the greatest common divisor of the coefficients of

X is 1.

2. Kronecker[105] called the rational integer a the norm of f and when a = 1 , the form f is called a unit form.

DEFINITION III-3. Two forms f and g are said to be equivalent in a restricted sense, written

f=g, when their quotient is equal to the quotient of two unit forms.

Remark

Every unit form f is equivalent to 1, that is,

f = l.

In other words, a form whose norm is 1 is an algebraic unit.

DEFINITION III-4. A form h iS divisible by a form f if there exists a form g such that

h sfg.

DEFINITION III-5. A form p is called a prime form if p, in a restricted senseof equivalence, is divisible by no form other than itself and unity.

One of the most fundamental concepts in the theory of forms is

that of content.

DEFINITION III- 6 . Let f be a form with coefficients a^, a 2 « . . .* a^; then the greatest common divisor of these coefficients is called the content of f and is denoted by ct(f), where the a^^ are elements of a fixed ground field k where g.c.d's exist. 114

DEFINITION III-7. An element

a = f(xi> ^ 2 , . . ■

g(xi, % 2 , . . of k(xi, X2 » . . .)Xy) is said to be an integral element or an integer if the g.c.d. of the coefficients of f is divisible by the g.c.d. of the coefficients of g.

Remark

The definition of an integer will be stated in terms of the con­ tent of a form later in this section.

Kronecker[105], in view of the above, defined his divisors

(ideals) as follows:

DEFINITION III-8 . Any finite sequence of integers a^, a2 , . . , a^ in a ground field k which do not all vanish determines a divisor,

a — (a^5 ^2, . . . ^a^).

DEFINITION III-9. An integer a in k is said to be divisible by a if and only if

a . . . +a^x^

is an Integral element of k(xj^, %2 , . . . x^), where a^x^^+agX2 + . . . a^x^sa.

Remarks

1. To each form of a particular field there corresponds a definite ideal

a — (a^, a2 , * . . ^aj^) in the ring of integers of that field.

2. To every ideal there corresponds anindefinite number of forms such as 115

f(xj^, x^, . . . * • * ^r^r* where the y^'s consist of any product of powers of x^, x^, , , .^x^ and y^ f y^.

3. Hilbert[8 6 ], whom we shall discuss later in this chapter, called the divisor (module) a the content of the form.

If we compare Dedekind's definition and Kronecker's definition of divisor, there is an obvious resemblance: Dedekind's definition becomes

Kronecker's definition upon the substitution of the indeterminates x^^, Xg, . . . jx^ of Kronecker for the arbitrary integers of Dedekind and the substitution of the divisibility of a by a^x^+SgXgf

. . . a^x^ by the equation

a = . . . +a^t^

(see DEFINITION 11-13).

Continuing with the concepts pertaining to forms, the theorem which follows demonstrates a very significant property of the content of a form. This theorem points out how the content of a form plays the role of a divisor. We give the proof of Weyl[215] for one indeterminate x^ so that the notation does not become unwieldy.

THEOREM III-l. Let f and g be forms of indeterminates x^, x^,

. . . , then

ct(fg) = ct(f)'Ct(g).

Proof:

Let a = ct(f), 6 = ct(g) and y = ct(fg). Let p be a prime number and let q be a number exactly divisible by p^; that is,

p^|q but p^'^^lq. 116

We must show that y is exactly divisible by that is to say, (A):

If f and g are exactly divisible by p^ and p^, respectively, then fg is exactly divisible by Suppose we order f and g by decreasing powers of x^; that is,

f(xj^, X2 » . . .) = ^0 ^ 1 ‘*'^1 ^ 1 ^”^'*'^2 ^1 ^ . . .

/ \ L ^ IT- m—1 . m-2, gCx^, X2 ; • . .) = bQX^^+b^Xj^ +^2 ^ 1 + • • • .

Then not all of the coefficients a^ will be divisible by p^^^. Let a^ be the first coefficient in f not divisible by p^^^ and bg be the first coefficient in g not divisible by Then the coefficient (ab)^^g of xj"^® in fg will be a form P such that

P 5 3 j.bg (mod p^+b+1^ ^

IThen the statement (A) (above) holds true for forms of one indetermin­ ate less, we can apply it to the two forms fj. and of X£ and find fj.gg. Hence the coefficient of (f6 ) ^ + 3 is exactly divisible by

Thus by induction on the number of indeterminates x^, X2 , . . . Xj., the statement (A) is true, that is

ct(fg) = ct(f)'ct(g).

Though a form, in general, is an integral function, the concept of content may be extended to rational functions of indeterminates

Xf, X2 » . . . via fractional divisors (content). ^ is used to denote the pair of integral divisors a and g with the convention

B “ g' whenever ag' = a'3. This equality is an equivalence relation. Multi­ plication is defined by 117

£L!_ = cto* g ' B' SB' and its result (product) does not change when one replaces either of the two factors by an equivalent one.

Further, we shall interpret the equation

f “ ^ to mean that

a = By­

definition 111-9. ^ is said to be integral or an integer if and only if a is divisible by B; that is, there exists an integral divisor

Y such that

a = By - f(x^» ^2 » • • •) DEFINITION III-IO. Let h(xn, x«, . . .) = - 7 — — ------r be J- i *2 * - - -'

an element of k(%i, X 2 , . . where f and g are forms of a fixed field k. Then the content of h is defined by

Ct(h) = • Ct(g)

Remarks

1. This definition defines the content of all rational func­ tions of indeterminates x^,......

2. One can observe that plays the same role as does the Ct(g) fractional divisor — , where a = Ct(f) and B = Ct(g). B

3. The above definition is consistent with the preceding one since

g hg = f = > 118

Ct(hg) = Ct(f)

But

Ct(gh) = Ct(h)*Ct(g) = 1 >

Ct(h)-Ct(g) = Ct(£) = >

In Section 1 of this chapter we mentioned the fact that Dedekind

used Q, the. field of rational numbers, as his starting point and that

Kronecker used a ground field k and a finite extension L over k, and

observed the invariant properties of the rational functions in moving

from kCx^, X 2 , . . .) to LCx^, X 2 * . . .) and vice versa. Since we

know that the integers of Q have a ring structure, we should like to

know whether or not Kronecker*s integers have a ring structure. Since

both theories thus far have been equivalent, we should expect an

affirmative answer.

DEFINITION III-ll. An element fCx^, X2 , . . .) of kCx^, % 2 , . . .)

is said to be an Integral element or an integer if its content is an

integral element of the fixed field k.

THEOREM III-2. The integral elements of k(xj^, X2 , . . .) have a

ring structure.

Proof:

Let f ^ 0, f* ^ 0 be elements of kCx^j ^2 , . . .) such that f and I f are exactly divisible by p^ and p^ respectively and similarly let

g # 0, g' f 0 be elements of kCx^, X2 > • • •) such that g and g' are

exactly divisible by p^ and p^ respectively, where a % b, a* - b*. Then

observe that 119

f + 1 1 = = . > g g' gg’

l + l l = paq.pb'r'+pa'q' -pbr S s' p^r-p^ r ’

f . f ’ p^'^'b \gr ' +p^ ' - g ' r

Hence the numerator p^"^^ *gr'+p^ ‘^'^•g'r is divisible by p^jwhere c = min(a+b*, a*+b) and the denominator is divisible by where d = b+b'.

Note that a - b and a' - b ’ = > a+b' - b+b^ and a'+b - b'+b. There­ fore, the sum of two integers in k(x^, X2 , . . .) is an integer in k(x^, Xg, . . .). Similarly the difference of two integers is an integer. Now

f f' = f f g ' 8 ' gg'

l . l l = 2Îpf_\ = = > g g' pbpb'

a+a' jL . H = £ g'g' pb+b'

Hence the numerator is divisible by p^ , where c' = a+a' and the de­ nominator is divisible by p*^. Therefore, the product of two integers

in k(x^, X2 , . . •) is an integer in k(x^, X 2 » • • •)• Thus the integral elements of kCx^, X2 > • •’•) have a ring structure.

Some additional definitions related to integral elements are required.

DEFINITION III-12. An integral element a of k(xi, X2 , . . .) is called a unit if both a and •- are integral elements. _ _ a 120

DEFINITION III-13. Two forms fCx^, • • -) and gCx^, xg*

. . .) are called associates, written

f~S, if 1 is a unit. S Remarks

1. An equivalent condition for f ^ g is that — and & are both ® f integral elements for k(x]^, X2 , . . .).

2. f'V g is a necessary and sufficient condition for f and g to have the same content.

Suppose k is the ground field of our discussion and that L is a finite extension field over k; also suppose that the ring of integers in k and the divisors in k satisfy the properties A, B, C, D, E,and F of Dedekind's divisors (Chapter II, Section 7). Kronecker[105] was successful in extending these concepts to the extension field L over k.

Let k(x^, X2 » . . .) and L(x^, Xg, , . .) be fields arising from k and L respectively by the adjunction of indeterminates x^, %2' ' ' ' '

We shall include the lowest case where n = 0. The following definition of integral element is applicable to the extension field L.

DEFINITION III-14. a is said to be an integral element of

L(x^, X2 » . . .) if it satisfies an equation

a’^+a]^a^“‘^+a2a^’"^+ . . .a^ = 0 whose coefficients are integral elements of k(x^, %2 , . . .).

Remark

The same argument used to show that the integral elements of k(x%, X2 » . . .) form a ring is sufficient to show that the integral elements of L^x^, ^ 2 , . . .) have a ring structure. 121

It is interesting to note that we can utilize the same defini­ tions cited in Chapter II, Section 7, that Dedekind[33] used in order to discuss the results that Kronecker[105] obtained in the ring

L(xj, , . . .). Some of these results, compiled by Weyl[215], are listed below,

DEFINITION III-15. An Integer a is said to be divisible by the integer b f 0 if ^ is integral, b DEFINITION III-16. a is associated with b , written

a "V b, means that a:b and b:a.

DEFINITION III-17. a f 0 is a unit if both a and — a are integers. Remarks

1. a'vb and a'^b' imply that aa'^bb'.

2. In general a ~b and a' ~b* do not imply that a+a'~b+b' .

3. An integer a in LCx^, ^2 , . . .) defines a divisor in L.

4. Integers a, b in L(x^, X 2 > . . .) define the same divisor in

L if and only if a'vb.

THEOREM III-3, If a is an integral element of L(x]_, X2 , . . .),

then its irreducible equation and hence its field equation in k(x^, Xg, . . .) have Integral coefficients.

Another important theorem, again stated without proof, is

THEOREM III-4, An element of kCx^, « 2 , . . .) which is integral

in LCx^, X2 , . . .) is an integral element of kCx^, X 2 * • • •)•

Remarks

1. Theorem III-4 permits the statement of the following crite­ rion for integrity in L(xi, X 2 , . . .): a is an integral element in 122

L(x^» X 2 » . . .) if and only if the norm (see Appendix A) of t-a, written N(t-a), is an integral element of k(t, X 2 » ■ * -) an indeterminate t.

2. The norm N(a) and the trace S(a) (see Appendix A) of an integer a in LCx^, X 2 , . . .) are integral elements of k(xi, X2 , . . .)•

DEFINITION III-18. The divisor norm of an element a, written n(a), is defined to be the content of the norm of a, that is,

n(a) = Ct(N

Remarks

1. n(ab) = n(a)n(b), where a and b are integers in L(xi, %2'

2, If a is a unit, then n(a) is the unit divisor in k.

In Section 1 of this chapter, we cited some of Kronecker's notions on divisibility in comparison with those of Dedekind. We shall return to these notions and their relation to the theory of forms. In so doing, we shall see that Kronecker's theory of divisors is the same as Dedekind’s theory of divisors as discussed in Chapter II, Section 7.

We shall follow a summary which was given by Weyl[2l5],

DEFINITION III-19. Suppose ct = (a^, 0 2 , . . • a^) and B = (b^, b2 > • • • bj.) are two divisors. Then ct = 6 if every integer divisible by ct is divisible by B and every integer divisible by B is divisible by ct. Hence a = B if n :B and B:ct.

Remarks

1. a is divisible by a if a is divisible by the linear form a-j^x^+a2 X2 + . . . +a^x^, where in this sense the linear form may stand for the divisor (a^, a2 , . . . a^). 123

2. Whether or not a is divisible by ct can be determined provided we are able to decide in the ground field k whether or not the gcd of certain numbers (a^, ag, . . .) is divisible by the gcd of certain other numbers (b^, b^, . . . ).

THEOREM XII-5. Each of the numbers a^, . . .,a^ is divisible by

^ ^2’ • • • *

Proof: *iyi**2y2+ • • • +*r?r We must show that r :------1 ---- for a^Xj^+a2%2+ • • • y^ = 1, Yg = y^ = . . . = y^ = 0 is an integer in L(x^^, x^, . . .jX^).

With the homogeneous form

f(x^, Xg, . . . Xp) = N(a^^x^+ . . . +a^x^) in k of degree n, we obtain as the field equation of

^l^l‘‘‘®2^2‘*‘ • * • ■‘■^r^r ((tXi-yj, Cx^-y^, . . . tx^-y^) a^x^+a2X2+ . . . +a^x^ " f(x^, Xg, . . . x^)

It is clear that the gcd of the numerator is divisible by the gcd of the denominator. Therefore each of the numbers a^, ag, . . .^a^ is divisible by a = (a^, ag, . . • •

THEOREM III-6. a is divisible by B if and only if

^l*l"^^2^2^ • . • +a^x^ 1^*1' *2- • ■

3^1» 3^2* * ' ‘ *^s^ * Proof: = >

Suppose a is divisible by g. Then a^:a implies a^:g, that is,

b^y^+b2 y;+ A ■ +b;y, Integers in bCy^. y,. . . . ,y,). Suppose

a^x^+a2X2+ . . . +a^x^ we multiply by x. and upon adding, we find that T— . t ------„ -L 1^1 2^2 * • • ’’s^s 124

Proof: < =

Suppose h y,+b,y,+ ...by integral In L(u^, Xj, . . . x^, 1 1 2 Z s s y^, Yg, . . . and a Is divisible by a, so that

a^x^+a^X;+ 7 : .-+a;3 Ç ^ integral In L(x^, Xj, . . . and hence in

L(Xj^, Xj, . . . Xj., yi, - - - y^)- Hence b^y^+b^yg+ . . . +b^y^

integral in L(x^, x^, . . . x^, y^, yg, • • • y^) and omitting the un­ necessary variables x^, Xg, . . . ,x^, we see that y ^ y ^ ~ 1 1 2 2 s s is integral in L(y^, y^, . . . Yg)• Hence a is divisible by B.

Remark

A criterion for the divisibility of a by g is Ct NCa^x^^+agXgf

. . . a^x^+b^y3^+b2y2+ • • - + V s ^ ' ^ ^ ^<^1^1+ ' ' * + V s ) '

THEOREM III-7. a = (a^^ Sg, . . . a^) is the greatest common

divisor of a^, a^» . . -a^.

Proof;

If B = (b^, bg, . . .jbg) is a common divisor of a^, Sg, . . •

then ^i b,y,.b,y,. . . . +b,y, . . . a^x,

is Integral in L(y^, y^, ... ) for each i and b,y,+b,y,+ ...by s

is integral in L(Xj^, Xg» » « « y^* 3^2* • • • Yg) » that is, ot.B* Hence

a is the greatest common divisor of a^, Sg, . . •

THEOREM III-8. Two divisors a = (a^, Sg, . . . a^),

B = (be, b„, . . . b ) have a greatest common divisor, namely

(a, B) ~ (a^^, Sg, . . • *a^, b^, bg, . . • ,ibg)* 125

Proof;

By the same argument given in Theorem XII-6 above, we see that

aj^x^+a2X2+ . . . +a^x^+b^y{+ b 2 Y • • • +bgyg a^xj^+a2X2+ . . . fa^x^+b^y^+b2Y2"*" * • • +bgyg is an Integer with y{ " y^ ” • • • = y^ = 0; that is, a is divisible by a^^Xj^+a2X2^ # * * (a,B) = (ag, &2 , . . . a^, b^, b2 , . . . bg). Now if : . .' arXr b^^y^^+bgygf . . .+bgyg are integral in LCx^, X 2 , . . • ÿ'i» V 2 »

) and L(y^, yg, . . . z^, Z2 » . . .) respectively, then their sum is integral in L(x^, Xg, . . . y 2 » . . . z^, Z2 , . . .); that is, if a and 3 are divisible by y» then (a,3) = (a^^ a2 , . . . ^1’ ^2»

. . . bg) is divisible by y.

We recall that a divisor (a^, Sg, . . •^a^) which can be expressed as aj^x-]^+a2 X2 + . . . +a^x^ is called a linear form. Now if a = (ajL, S2 , . . .) and 3 = (b^, b 2 , . . .) are two divisors then their product appears to go beyond the bounds of linear forms because the product of two linear forms aj^xj^+a2 X2 + . . . fa^x^. and bj^xj^+b2 X2 + . . . bgyg yields a bilinear form Ea^b^Xj^y^ (i = 1, . . . r, k = 1, . . , s).

However, the product of two divisors does not go beyond the bounds of linear forms because

(EaiXi) (Ebfcyi^) ~ Za^b^u^^^ with rs indeterminants u^^y

DEFINITION III-20. An integer c is divisible by u3 if

is integral in L(x, y). EaiXi*Ebi,yk 126

Another very important relation between the divisors in k and L

is established by forming the norm; the divisor in k,

d = N.(a2^x-j^+a2 X 2 + . . . 4-a^x^) = CtN{a^Xi+a2 ^ 2 + ■ • • +aj»Xj.)

is said to be the norm of a = (a^, 3 2 , . . . a^). Finally we conclude our discussion of Kronecker's forms by stating, without proof, a

theorem which is most germane to our discussion.

THEOREM III-9. The norm of a form f is the norm of its content,

that is,

N(f) = NCt(f).

In summary, again, we can say that both Dedekind and Kronecker were attempting to find a general arithmetic of algebraic number fields when they arrived at their respective ideals and divisors. In both

cases they distinguished a set of elements which they called integers

from a set of elements which they called ideals and divisors respec­

tively. Their ideals and divisors have the same properties it so happened. Kronecker's theory, however, was the one which was used by

the algebraic geometers.

Suppose we let C be the field of complex numbers. The elements of C(xj^, X 2 » . . . x^) are the rational functions of m > 1 variables or indeterminates x^, X 2 , . . . x^ with arbitrary complex coefficients.

It can be shown, as above, that the integers (= the divisors of C) are the polynomials of X]_, X2 , . . . Xg^ which form the ring C(xj^, X2 , . . .

Xj„) . For example, let m = 3.

A single polynomial equation

f(x]^, X2 * X3 ) = 0 defines an algebraic surface in the three-dimensional space E3 (see 127

Appendix B) with coordinates x-j^, X2 , x^. The law of unique factoriza­

tion states that any such algebraic structure splits in a unique manner

into irreducible surfaces, By passing from k to an extension field L

over k of finite dimension n, we pass from the space Eg to an algebraic

3-space F g of degree n over E g . Kronecker's theory tells one how to

define algebraic surfaces in F g in such a manner that the law of

unique factorization into irreducible components still prevails. In his theory, the divisors represent algebraic surfaces; a;6 is inter­

preted in geometrical language as "the surface 3 is part of a"; the

product aB is interpreted as the "union of the two surfaces a and B-"

Hence Kronecker's theory actually amounts to a very reasonable geometry

of surfaces in F g . This, of course, was not true of Dedekind's theory

since it can be shown that there is no preservation of unique factori­

zation in passing from Q to C(X). We shall discuss more of these

algebraic-geometrical concepts in Section 6 of this chapter.

As we can see from the above, Kronecker's theory of forms in­

volved an ideal theory which paralleled that of Dedekind, In fact, the

two theories are actually equivalent. In view of this fact, one is most likely to be concerned about which of the two theories is more

convenient. Though this is still somewhat debatable today, Weyl[215l

favored Kronecker's theory. He contended that Dedekind's theory lacked

a certain type of self sufficiency in that some of his proofs employed

the indeterminate Ideals of Kronecker's theory of forms. We shall see

in Chapter III that Emmy Noether favored Dedekind's approach. 128

3. Hilbert and His Basis Theorem

David Hilbert (1862-1943) was one of the world's greatest mathematicians. According to Gillispie[60], he was descended from a

Protestant middle-class family that had settled in the seventeenth century near Freiberg, Saxony. After receiving his Ph.D. in 1885 from

the University of Konigsberg, he was appointed to a chair at Gottingen

in 1895, remaining there until his official retirement in 1930. During his tenure at Gottingen, this university became one of the greatest mathematics centers of the world.

Hilbert greatly influenced the development of American mathemati­ cians. Between 1900-1914, many young American mathematicians migrated

to Gottingen to study with him. Hilbert's research activities can be divided into six periods according to the years of publication of his results: (1) up through 1893, algebraic forms ; (2) 1894-1899, algebraic number theory; (3) 1899-1903, foundations of geometry; (4)

1904-1909, analysis; (5) 1912-1914, theoretical physics; and (6)

foundations of mathematics.

It is no coincidence that algebraic forms were his first area of

interest. The mathematician whose work most greatly influenced Hilbert was the great number theorist, Kronecker. Hilbert's research on

algebraic forms centered around certain invariant properties of these

forms which he called the theory of invariants.

Weyl[216] points out the fact that the classical theory of

invariants deals with polynomials f *» f (x^^, Xg, . . . x^) depending on

the coefficients x-j^, X2 > . . . x^ of one or several ground forms of a

given number of arguments D-j^,n2 , . . . hg. He asserts that any 129 linear substitution s of the g arguments with determinant 1 induces a certain linear transformation U(s) of the variable coefficients x^, x^,

. . . x^, X ^ x' = U(s)x, whereby f = f(x^, Xg, . . . x^) changes into a new form

f(x', x^, . . . x^) = f^(x^, Xg, . . • x^).

DEFINITION III-21, f is said to be invariant if f® = f for every s.

Remarks

1. The restriction to transformations s of determinant 1, which are called unimodular transformations, enables one to avoid thé more involved concept of relative invariants.

2. The restriction to unimodular transformations s removes the restriction to homogeneous transformations; this causes the set of such invariants to have a ring structure.

The classical problem is a special case of the general problem of invariants in which s ranges over an arbitrary given abstract group G and S U(s) is any representation of that group, that is, a law according to which every element s of G induces a linear transformation

U(s) of the n variables x^, x^, . . , , x^ such that the composition of group elements is reflected in the composition of induced tranformations.

The development of invariant theory prior to Hilbert had produced two main theorems which, however, had only been proven in very special cases.

The first theorem states that the set of invariants has a finite integral basis; that is, we can pick a finite number among them, say if, ig, . . . i^ such that every Invariant f is expressible as a 130

polynomial in i^, ig, • • The second main theorem asserts that

the relations have a finite ideal basis; that is, one can pick a

finite number among them, say Fj^, F2 , . .. ,F^, such that every relation

F is expressible in the form

F =

the being polynomials in the variables z^, Weyl[216]

points out the fact that an identical relation between the basic

Invariants i]^, i2 » . . . ,i^ is a polynomial F(z]^, Z2 , . . • *Zni^ of m

independent variables z^, Z2 > * * **^m ^bich vanishes identically by

virtue of the substitution.

zjL “ %2, . . . , Z2 — ^2’ * " * » - »

^m = imt^l* • ' ' j V '

Hilbert[8 6 ] proved the second of these two main theorems of invariant

theory.

The relations F form a subset within the ring k[z^, Z2 , . . .$z^]

of all polynomials of z^, Z2 , • • .,z^ the coefficients of which lie

in a given field k. When Hilbert found his simple proof of the second main theorem, he could not fail to note that it applied to any set of

polynomials Z whatsoever and thus he discovered one of the most funda­ mental theorems of algebra which was instrumental in ushering in our modern abstract approach to algebra, that is:

THEOREM III-IO. Every subset of the polynomial ring k[zj^, Zg,

. . • jZjjj] has a finite ideal basis.

This theorem, known as Hilbert's Basis Theorem, says that given

a ring of invariants, then any invariant f can be presented in the form

f = Qifi+q2f2'*‘ • • ■ ‘^k^k* 131 where Qg, • • • are polynomials which may be assumed to be of lower degree than f. In more modern terms, this theorem states that if R is a ring such that every ideal has a finite basis, then its polynomial ring R[x] is such that every ideal has a finite basis. This theorem is so significant that we shall state and prove it in its original form and in a more modern form. It had been a practice of mathematicians during that era to consider infinite collections of polynomials called modules. Hence some definitions are in order. We shall follow the definitions and proofs given by Macaulay[142] who, as we shall see in Section 7, was greatly attracted to the works of

Hilbert.

DEFINITION III-22. A module M is an infinite collection of polynomials or forms of n variables x^^, Xg, . . , ,x^ defined by the property that if F, Fj^, F2 belong to the collection, then F2 +F2 and AF also belong to the collection, where A is any polynomial in X]^, Xg,

• • •

DEFINITION .111-23. A basis of a module M is any set of members

Fi, F2 , . . • .Fk such that every member of M is uniquely expressible in the form Xj^Fj^+X2 F2 + . . . +FkKjp where X^, Xg, . . . X^ are polynomials.

The principal set of members of a module M is a generating set closely related to the basis of a module. We give its definition.

DEFINITION III-24. If a module M = (Fj^, ...... F^) , and no one of the polynomials F^, F2 , . . . is a member of the module whose basis consists of the rest, then F^^, F2 , . . . F^ is called a principal set of members of M. 132

DEFINITION III-25. An H-module (Hilbert-module) is a module hav­ ing a basis whose members are all homogeneous polynomials not neces­ sarily of the same degree.

DEFINITION III-26. A K-module (Kronecker-module) is a module, which in general, does not have a basis whose members are all homogeneous ploynomials.

Remark

An H-module M is said to be equivalent to the K-module =1, n whose basis is obtained from the basis of M by putting = 1 if each member of i becomes a member of M on simply reinserting x^ so as to make it homogeneous without altering its degree.

DEFINITION III-27. A module is said to contain a point a^^, 3 2 »

. . . if all of its members vanish when x^ = a^^, X 2 = 8 2 » • • • x^ = a^. (The only module which contains no point is the one which has every polynomial for a member.)

DEFINITION III-28. A polynomial of degree 5, is said to be regular or irregular in ,according to whether or not the term is present in it.

DEFINITION III-29. An elementary member of the module (F^, ^2 »

. . . jFj^) is any member of the type wF^(i = 1 , 2, . . . .n), where w is any power product of x^^, X 2 , . . .

Remark

An elementary member depends on the basis chosen for the module.

DEFINITION III-30. DD^^) ^ is called the complete

(total) resolvent of the equations F^ = F2 = . . . = F^ = 0 and of the module (Fj_, F2 , . . . F^), where D is the H.C.F. of F^, F2 , • . . 133

D (1) is the H.C.F. of F ^ \ F^^) . . . is the H.C.F. of F p \ F^^\ . . . ,F^^) ; and the H.C.F. of F^’^"^) ^ . . . ^

and F^^) ^ . . . jF^^) are polynomials in Xj, Xg, . . .

F^^\ ^2^^* • * * are polynomials in xg, x^, . . . x^; . . . ';

are polynomials in x^_^, x^.

Remark

jj(i-l) called a complete partial resolvent of rank i and any whole factor of ^(^"3.) ^g called a partial resolvent of rank i and any poly­ nomial factor of is called a, complete partial resolvent of rank 1,

DEFINITION III-31. A module F is defined to be simple or mixed according to whether or not one or more than one of the complete partial resolvents D, differ from unity.

DEFINITION III-32. The complete resolvent D^D^^^)^

= F^ of (f^, f2 » • • • ;fk) obtained by eliminating X 2 , xg, . . . jx^ in succession is called the complete u-resolvent of (Fj^, Fg, . . . ^F^) ,

DEFINITION III-33. The rank of a module (more especially of a primary module) is the least number of the variables which must be functions of the rest in order that all the members of the module may vanish; and the number of the remaining or arbitrary variables is the manifoldness or dimensionality of the module. (The modern definition asserts that the rank of module is the number of basis elements.)

DEFINITION III-34. The spread or variety of a module is the totality of points which it contains.

Remark

Let F-j^ = F2 = . . . = Fk = 0 be a system of equations in n un­ knowns x^, Xg, . . . Xjj, A set of solutions for x^, X g , . . .jX^. when 134

. . . x^ have arbitrary values is said to be of rank r and the spread of the points, whose coordinates are the solutions, is of rank r and dimension n-r.

DEFINITION III-35. Any member F of a module M is said to contain

M.

Remarks

1. The term "contains" is used to denote an extension and generalization of the phrase "is divisible by."

2. In more general terms, one can say that a module M contains

M' if every member of M contains M'. (This will be the case if every member of the basis of M contains M'. ) Thus (Fj^, Fg, . . . F^) con­ tains (Fj^, Fg, . . . F^, Fk+]^) and thus a module becomes less by adding new members to it, (See Appendix A.)

3. There is a module which is contained in all modules, it is called the unit module and denoted (1).

4. Though the concept of "contain" is not the usual term used in the discussion of modules, we shall, however, maintain its use as well as some of the other obsolete terms because many of the earlier forms of the theorems are expressed in terms of them. (A glossary will be given later in Section 5 of this chapter.)

DEFINITION III-36. The least module of a finite or infinite set of modules is a module which contains every other module.

DEFINITION III-37, The G.C.M. (greatest contained module) of k given modules M^, Mg, . . . is the greatest of all modules M con­ tained in Mj^, Mg, . . . ,Mk and is denoted by (M^^, Mg, . . . M^) . 135

DEFINITION III-38. The L.C.M. of Mg, . . . .M^ is the least of all modules M containing M^ and Mg, . . .*M^ and is denoted by

[Mj^, Mg, . « « jM^] .

DEFINITION III-39. The product of M^, Mg, . . , jM^ is the module whose basis consists of all products Fj^Fg . . . F^, where Fj_ is any member of the basis of (i = 1, 2, . . .,k). The product is denoted by M^Mg . . . M^ and is evidently a module independent of what bases may be chosen for M^, Mg, . . . .M^*

Remark

The product M^Mg. . . M^ contains the L.C.M, [M^, Mg, , . . M^]»

DEFINITION III-40. The residual of a given module M' with

respect to another module M is the least module whose product with M*

contains M and is denoted by .

Remark

The members of the residual consist of every polynomial whose

product with each member separately of the basis of M' is a member of

M.

The following remarks are pertinent to the general theory of modular systems (modules):

Remarks

1. There are two theories of modular systmes: the algebraic or

relative theory and the more difficult and varied absolute theory.

2. In the algebraic theory, polynomials such as F and aF, where

a is a quantity not involving the variables, are not regarded as dif­

ferent polynomials and any polynomial of degree zero is equivalent to 1. 136

3. In the absolute theory, the coefficients of F^, Fg, . . . are restricted to an integral domain which is generally the ordinary integers or whole functions.

4. We shall denote the fact that F is a member of M in the relative or absolute sense by F = 0 mod M or F = 0 mod M respectively, where F is any polynomial belonging to a module M.

The following theorem is a reproduction of the Basis Theorem of

Hilbert[86] which has been analyzed by Macaulay[142].

THEOREM III-ll. If F^, F^, , . , is an Infinite series of homogeneous polynomials, there exists a finite number k such that

F^ = 0 mod (F^, Fg, . . . when h > k.

Proof:

It is clear that F^, Fg, . . . are given in a definite order.

In the case of a single variable, the series F^, Fg, . . . consists of powers of the variable and if F^ is the least power, then

F^j = 0 mod F^ when h > k. Hence the theorem is true in this case. We now assume it for n-1 variables and prove it for n variables.

The series F^, Fg, F^ . . .i s called a modifiedform of the series F^, Fg, , , . if

Fi = F^ and

Fi = F^mod(F^, Fg, . . . for i > 1, Thus the modules (F^, Fg, , . . Fj,) and (Fi, Fg, . , , ,Fi) are the same. The theorem will be proved if we show that the series 137

F^, Fg, . . can be so chosen that all its terms after a certain number become zero. We assume that Fj^ is regular in and we choose the modified series so that each of its terms F^ after the first is of as low degree as possible in x^, and therefore of lower degree in x^ than Fj^. The terms of the series F^, F^, . . .o f degree zero in will be polynomials in Xj^, Xg, . . . andthese can be modified so that all after a certain finite number become zero, since the theorem is assumed true for n-1 variables. LetFj , Fl , Fj, . . . be all the 1 2 3 terms of F^, F^, F^, . . . taken in order, which are of the same degree & > 0 in x^ and let f^^, f^^, . . . be the polynomial coefficients of x„ in them. Then f.', f.*, f.», . . . are polynomials in n-1 1 2 3 variables; and we cannot have

f ' = OmodCf’ , f^ , . . . jf^ ) ^i ^1 2 i-1 for any value of i for if

then F' -A,F' . . . -A.F is of less degree than H in x , which *i *1 1-1 cannot be. Hence the number of polynomials f, , f, , . . . or the 1 2 number of terms Fix-i , Fj , . , . in the series is finite. Hence the number of values of I is also finite, the greatest value of & being the value it has in F^. Hence there exists a finite number k such that

F^= Oraod(FjL, Fg, . . . ^F^) when h k k. 138

Corollary. Any module of polynomials has a basis consisting of a finite number of members.

Among several equivalent definitions (see Appendix A) a

Noetherian ring can be defined as below.

DEFINITION III-41. A Noetherian ring is a commutative ring R such that every ideal of R is finitely generated.

This definition is necessary for the statement and proof of the more modern form of Hilbert's theorem.

THEOREM III-12 (Hilbert's Basis Theorem). If R is a Noetherian ring, then the polynomial ring R[x] is also Noetherian.

Proof !

We shall suppose that I is an ideal of R[x] and we shall show that I is finitely generated. The elements of I are polynomials. We form a set of elements of R by taking the leading coefficients of all the polynomials in I together with the zero element. Suppose that jl» then there exist polynomials j]^x^+ . . . and jgx”+ . . . which are in J^, where in both cases + . . , stands for : plus terms of lower degree. Put p =■ m+n, multiply the first polynomial by x^ and che second by x*"'. Thus we obtain two polynomials j^x^d- . . . and jgxP+ . . . both of which are in I, hence (j2 %^+ • ■ .)±(j • * •) el, thus j-i^ijgeJ^. Again if reR, then r(jj^xP+ . . .)el and therefore rjj^eJ, Hence is an ideal of R. But R is Noetherian; this implies that is finitely generated, say

“ (ill j2> • • ' dh) so that there exist polynomials

= fl(x), fg = fg(%), . • • 139 such that has leading coefficient j^. Now by multiplying each of the f^Cx) by an appropriate power of x, we can arrange that they all have the same degree, say the degree N, thus

f^sl, f^(x) = j 1 %^+ . • • (l£i5h).

Next consider all the polynomials in I whose degrees do not exceed

N-1. The coefficients of x^“T in these polynomials form an ideal Jg of R, say

Jg = (k^, kg, . . . ,k^) and we can choose polynomials g^ (x), gg(x), . . . #gjjj(x) such that

g^el, gj^(x) = k^x^"T+ . . . (1-i-m),

In the same way, if we consider the -polynomials in I whose degrees do not exceed N-2, then the coefficients of in these polynomials form an ideal

Jg = , &g, . • U - •

Also there will exist polynomials h](x), hg(x), . . .^h^(x) such that

h^El, h^(x) = . . . (l^i-^n).

By this method we eventually obtain a certain finite set f^, fg, . . . fh* gg» • • • Em* hg, . . . ,h^ . . . of polynomials. These poly­ nomials are all in I. We shall now show that, in fact, they generate

I. Suppose 4>(x) = jxP . . . belongs to I, then jsJ^, say

j = ü)3^j]^-fmgjg+ . . . +Whjh' where w^sR. If p 2 N, then

xP‘'^fj|_-ti)2xF“^f2- . . . -w^xP^^fh is again in I, but it has a smaller degree than ifi. If the degree of the new polynomial is still not less than N, we can reduce it further 140 by the same device. In this way we see that there exist polynomials

Aj^Cx), Ag(x), . , .jAjjCx) such that

4)(x) = A]^(x)f^(x)+Ag(x)f2(x)+ . . . +AkCx)fk(x)+i|i(x), where ii'(x)el, and N-1. is the degree of ij».) The proof will be complete if we can show that

!|;Cx) = Wi8i(x)+ . . . "f^lbl( + • • • +V£hjj(x)+ . . . , where ^l> • • • '’Jl • • • are all in R. To do this, first

choose P^jPg» • • • Mk that iji(x) and P]^gj^Cx)+Pgg2 (x)+ . . . +Pkgk(x) have the same coefficient of x^“T. it is clear that this is possible

since iij(x)el and N-1, by the definition of the gj_(x). Next choose

Vg, . . . Vjj^ so that the coefficient of x^"^ is the same in

4)(x)-pj^g^(x)- . . . -WkEk(^) as it is in Vih2 (x)+Vghg(x)+ . . . Vg^hg^(x).

Continuing this way we obtain

4f(x) = PlEi(^)+ • • • '*'bk6k(^)'^^lbl • • • +V£hg^(x)+ ....

Hence R[x] is Noetherian.

Corollary 1. If R is a Noetherian ring, then the polynomial ring R[xi, Xg, . . . ,x^] is also Noetherian.

Proof;

Put Rq = R, and R^ = [xi, Xg, . . . for 1 5 i S n. Since every polynomial in x^, Xg, . . . can be regarded in precisely one way as a polynomial in x^_^^ whose coefficients are polynomials in xi, Xg,

. . . jXj^, the ring R^^]^ is none other than the polynomial ring.

Ri[xi^l]. Accordingly, by the above theorem Rj^^i is Noetherian when­ ever R^ is Noetherian . But by.hypothesis Rg is Noetherian, conse­ quently all of the R^ are Noetherian and in particular this is true for Rjij — R[Xj^, Xg, . . . Xj^]. 141

Corollary 2. If F is a field, then the polynomial ring

F[xi, Xg, • • • , x^] is Noetherian.

Hilbert's Basis Theorem is very essential to the foundations of modem algebraic geometry. Since each f(x^, Xg, . . . , x^) represents an algebraic surface in n-diraensional space, then all the information acquired about Noetherian rings is applicable to the Noetherian rings R^Lx-j^, Xg, . , . , x^]. We shall see in Chapter V that modern algebraic geometry is actually commutative ring theory and that the polynomial rings are the rings that are primarily studied in this theory. (If R is a commutative ring, then R^x^, Xg...... x^] is a commutative ring.)

Weyl[216] asserts that Hilbert's theorem is the foundation of the general theory of a mathematical structure called an algebraic manifold. (Algebraic manifolds will be discussed in a more detailed manner in Section 6 of this chapter.) Further, he contends that if we think of k more specifically as the field of all complex numbers, it seems natural to define an algebraic manifold in the space of n coordinates Xj^, Xg, , . . , by a number of simultaneous algebraic equations

fl = 0, fg = 0, . . . , f^ = 0, where the f^ekLx^^, Xg, . . . , x^]. (We shall see that a manifold is actually the set of common zeros of these f ) Now by Hilbert's theorem nothing would be gained by admitting an infinite number of equations.

Now if one denotes by Z(fj^, fg, ...» f^) the set of points

X — (xj^, Xg, ... , X^} , where f%, fg, , . . ,f^ and hence all elements of the ideal 142

a = (fi, f2, . . . i£^) vanish simultaneously. Then g vanishes on ZCf^, fg» * ' - when­

ever geCf^, f2 » • • • ,2%) but the converse is not generally true. For O example, vanishes whenever x^ does and yet x^^ is not of the form

2 xi*q(xi, % 2 , • • ' jXq)' The language of algebraic geometry distin­

guishes here between the simple plane x^ = 0 and the triple plane x^ = 0 although the set of points is the same in both cases. Hence in

algebraic geometry what is meant by an algebraic manifold is the

polynomial ideal and not the set of points of its zeros. But even if

one cannot expect that every polynomial g vanishing on

Z(fi, f2 , . • . *fh) = Z(a)

is contained in the ideal a = (f^, f,, . . . ,f^), one hopes that at

least some power of g will be. Hilbert’s "Nullstellensatz" states

that this is true if k is the field of complex numbers.

4. Hilbert’s Nullstellensatz

Also fundamental to algebraic geometry is Hilbert's Nullstellen­

satz.

THEOREM III-13. Let k be the field of complex numbers; if f is

a polynomial in k[x^, x^, . . . which vanishes at all the common

zeros of f2 » . . . ,ff in A^(L), then

fS = hj^fj^+h2 f2 + ■ * ‘ ^'^r^r

for some natural number g.

Before we carefully analyze and prove this convenient form of

the theorem, some preliminary ground work must be done in terms of

definitions and theorems. We shall use van der Waerden[205] as our

main source. 143

Remark

A^(L) Is the affine space over the field L (see Appendix B),

DEFINITION III-42. Let K be an arbitrary ground field with an extension L. L is called a universal field over K if L is algebrai­ cally closed and if L has infinite transcendency degree over K (see

Appendix A).

Remarks

1. In most works on classical algebraic geometry, the universal field is usually the field of complex numbers.

2. If K is given, the universal field can be constructed by first adjoining infinitely many indeterminates Xj^, x^, . . . to K and secondly by forming the algebraic closure.

The next theorem demonstrates the usefulness of the universal field. It shows that an isomorphism can be constructed between an extension field K(a^, . . .,0 ^^, a^eM 2 K and K(a^, a^, . . . , a^EL of the universal field which leaves the elements of the field K fixed and takes a^, Og, . . • into a|, otg, Ug, . . . ,a^.

THEOREM III-14. Any field extension K(a^, Og* ' ' ' obtained by adjoining finitely many field elements a^, Og* • * • to K can be isoraorphically embedded in L, the algebraic closure of K, where the

2 K.

Proof;

The elements a^, Og, . . . can be enumerated so that Og,

, . . are algebraically independent over K, while the other are algebraic over K(a^, Cg, • • • j«j.) • We now choose a^, . . ..^a^ algebraically independent over K in L. Then there exists an Isomorphism 144

( % 2 , « « « « 2 . ^2 * • • •

which leaves the elements of K fixed and takes a^, «g, . . . into

“2» • • • »’^r* if t = n, we have finished. If r

a zero of an irreducible polynomial d(x) with coefficients in

KCa-j^, . . . otj.) . To this polynomial there corresponds a polynomial

d'(x) with coefficients in K(aj, Og, . . . which has a zero

in K. Hence we can extend the isomorphism KCa^, «2» • • • -

KCo^, “2 » ’ • **®r^ to an isomorphism

KCa^, ^*2 » • * * >*^r+l^ ” ^^*1 * *^2 ’ ’ * * ***r+i^

which takes into Continuing in this manner, we obtain the

desired isomorphism

» * * * " ^^^1 ’ ^2* ’ * * g^n^ *

DEFINITION IXI-43. Let L be a universal field over a base field

K. Also let ÇeL. Then g is a point of the affine space A^(L) and Ç

is called a generic zero of an ideal P if feP = = > f(Ç) = 0 and con­ versely.

THEOREM III-15. If Ç-j_, ^2 * • • * >^n elements of an

arbitrary extension field of K, then the polynomials f in K[x]^, % 2 ,

. . . for which f(g) = 0 form a prime ideal in K[xj^, X 2 » . . . which is distinct from K[x^, X2 , . . . .

Proof:

Let f, geKtxj^, X2 , . . . such that f(Ç) = 0 and g(Ç) = 0

then f(ç)-g(Ç) = 0. Let heK[xi, X 2 , . . . . Now fCç) = 0 = >

f(C)h(Ç) = 0. Therefore the polynomials in question form an ideal J.

Since f(Ç)» gCC)» • • • are elements of a field, then J is a prime 145

Ideal and since J does not contain the identity, J is distinct from

K[x^, Xg, . . .

The following theorem proved by van der Waerden[205] character­ izes the unit ideal in the universal field L.

THEOREM III-16. Any ideal

a = • jfJ.) which has no zeros in L is the unit ideal.

Proof;

Suppose there exists an ideal a ^ u, the unit ideal, such that a has no zeros in L. By the Maximum Principle, there also exists a maximal ideal M f u without zeros in L. But a maximal ideal in a ring with identity is prime. Hence M is a prime ideal. But a prime ideal

M u has zeros. This is a contradiction since M # u is without zeros.

Therefore a = u, the unit ideal.

The above theorem can be stated in a form, given below, which is a special case of Hilbert’s Nullstellensatz. Rabinowitsch[175] proves this form of the famous theorem.

THEOREM III-17. If the polynomials f^, fg, . . . have no common zero in A^(l), then

1 ■ 8 1 ^1 + 8 2 *2 + • • • + 8 ^*r- Proof (THEOREM III-13):

For f = 0 the assertion is clear. In the case f 0 we adjoin a new variable z. The polynomials f^, f^, . . .yf^, 1-zf then have no common zero in A^^^^(L). By Rabinowitsch[175], it can be shown that 146

1 = eifi+82^2‘^ • * * +Sr^r'^S'Cl-zf) •

Now in this identity we make the substitution z = i . Hence

1 = 8 1 2 1 + 8 2 ^2 + • • • Now multiplying by a power fS to remove any resulting fractions we obtain

f® = hj^f j^+h2 £2 + • ■ • +h^f^.

The Nullstellensatz can be extended to the following theorem:

THEOREM III-18. If the polynomials p^, P2 > . , . .,Pg vanish at all the common zeros of f^, fg, . . . >f^, then there exists a natural number q such that all products of powers of the p^ of degree q belong

to the ideal (f^, £2 » • • • *

Proof

We observe that Hi p^ = oCfj^, f2 , • • • j£]-) •

We let

q = (qj^-l)+(q2 -l)+ • • • + ( q g - l ) + l .

Now every product of powers • • • P g ^ with h^t b2 + • • . h^ = g contains at least one factor pj_*^i; for otherwise h^^+h2 + . • • +hg would be at most equal to

(q2 ^-l)+(q2 “l)+ • • • +(qg~l) “ H "i-

Therefore q = (q]^-l)+(q2 -l)+ . . . +(qg-l)+l is a natural number such that all products of powers of the p^ of degree q belong to the ideal

(fj^> f2 * • • * *

Hilbert's Basis Theorem and Nullstellensatz were major contri­ butions to ideal theory and the general.theory of algebraic geometry.

It had been said by Gauss that number theory was the queen of 147 mathematics. Whether this is true or not is debatable; however, no other field of mathematics has attracted the elite of mathematicians with such an irresistable force as did number theory. Hilbert was a victim of this force. In spite of his outstanding work on the theory of invariants, he left this field of study to pursue the study of algebraic number theory. Lasker made a significant contribution to ideal theory after Hilbert's departure.

5. Lasker and His Primary Ideals

Emanuel Lasker was another one of the long line of eminent German mathematicians who studied ideals. According to Poggendorff[171], LAsker was born on December 24, 1868 in Brandenburg, Germany. He studied at the universities of Berlin. Gottingen and Heidelberg. Lasker, like

Hilbert and Kronecker,. realized the importance of polynomial rings and ideals and their relations to geometry. It had been established by the students of ideals in polynomial rings during that era that the main problem of polynomial ideals was to determine whether or not a poly­ nomial f belongs to a given ideal

M = (f^, fg, . . . fy).

This does not imply that a computational decision method was sought but rather a method which gives an insight into the structure of the ideal and expresses the relation between the zeros of the ideal and an element f. Lasker[132] was the-first to give such a method.

Lasker’s method depends on the decomposition of ideals into primary components. His method centers around a basic theorem which we shall state and give Lasker’s proof. However, some definitions are required at this point in our discussion. 148

DEFINITION III-44. A prime module is defined by the property that no product of two modules contains it without one of them containing it.

DEFINITION III-45. A primary module M is a module such that if the product of any two modules contains M, then one contains M or the other contains where is a prime module said to be determined by

M. (A primary module M is also defined to be a module whose spread is irreducible.)

Remarks

1. The product of the G.C.M. and L.C.M. of two modules contains the product of two modules.

2. The class of primary modules contains the class of prime modules.

DEFINITION III-46. Let R^ be any irreducible factor of F^, the complete resolvent; then corresponding to R^, as an irreducible spread,

is the spread of all points ^2 i’ ' ' *)^ri’ ^r+1 ' * * * in which . . . ,x^ take all finite values and Xj^^, Xgj^, . . . are the d sets of values supplied by the linear factors of R^ which vary

as Xj.^2 * * • * fSi

Remarks

1. It can be shown that any irreducible spread determines a prime module.

2. It can be shown that every primary module has a correspond­ ing prime module determined by its irreducible spread. 149

DEFINITION III-47. If M is a primary module and the corres­ ponding prime module, then the least number y such that contains M

is called the characteristic number of M.

DEFINITION III-48. A simple module is a module which contains only one point, say (M^),

Remark

In modern terms a module M over a ring R is simple if and only if

its only submodules are the zero module and the module M itself.

Now that we have established some of the basic definitions in their original form, we shall make use of Macaulay to state and prove

the theorem which was Lasker’s contribution to ideal theory.

THEORY III-19 (Lasker), Any module M is the L.C.M. of a finite number of primary modules.

Proof:

Let M be a module of rank r. Express its first complete partial u-resolvent in irreducible factors (see modern terms at the end of this section); that is,

and let C^, C2 , . . • denote the Irreducible spreads of dimensions n-r, corresponding to R^^, Rg, . . . Rj respectively.

Consider the whole aggregate Mj, of polynomials F for each of which there exists a polynomial F', not containing such that

FF’ = 0 mod M.

We shall prove first that is a primary module whose spread is

(i = 1 , 2 , . . . j j ) « 150 t Let F^, F2 be any two elements of Then since F^F]^ = 0 mod M, and F2 F 2 = 0 mod M, where neither F-[ nor F2 contains C^, we have

(A^F]^+A2 F 2 )F£F2 “ 0 mod M, where F^F^ does not contain C^. Hence A]^F^4 -A2 F2 belongs to the aggregate that is, is a module.

Again, since FF' = 0 mod M, F contains and Mj. contains C^.

Now, if Fy is the complete n-resolvent of M, then

(£'u^x = U]^xj^+U2U2+ . . . +Uq^%n " ^

while . U1 X1 +U2 X 2 + . . . +u„x„

(yu>u2 X2 + . . . "kieh eontalns Ci- Hence

= U2 %i+U2 X2 + . . . ~ ^ ^i'

But the polynomial (R^)^ = uj^X'j^-l-U2^2‘^ * ' * '^n’^n» not vanish identically, that is, irrespective of u^, U2 , • • • u^, for any point outside C^. Thus contains no point outside Cj^; that is, is the spread of

Lastly is primary; for if

F"F"' = 0 mod Mj^, then

F'F"F'" = 0 mod M, where F ’ does not contain C^; hence if F" does not contain C^, then

F'F" does not and F'" = 0 mod M^. Hence also if M"M'" contains and

M” does not contain C^, then M"' contains Mj^. Thus is a primary module whose spread is C^. Also M contains for every member of M is a member of N^. 151

The module M/M^ does not contain for if

M± = (^1^2* ' ' " and F^, F^, . , . ,F^ are polynomials not containing such that

Fj^F^ = 0 mod M (A = 1, 2, . . . jk), then

F^F^F^ . . . F^ = 0 mod M (5, = 1, 2, . . . k).

Hence F^F^ . . . F^ is a member of M/M^ not containing and there­ fore N/Mj^ cannot contain C^.

Since M/M^ does not contain C^, M/M2 , • . . ^M/Mj ) does not contain any of the spreads C2 , . . vjCj* We must prove that if is any single member of , M/M2 , . . . which does not contain any of the spreads C^, Cg, . . • jCj, then

M = [M^, Mg, . . . yMj, (M, (J>)].

Since M contains [M^, Mg, . . • *Mj, (M, 4>) ] » if has only to be proved that the latter contains M or that

F ■= 0 mod [Mj^, Mg, . . • jM., (M, ] requires F = 0 mod M. We have F = 0 mod (M, = f

f does not contain C^, then

f = 0 mod M^ = 0 mod [M^, Mg, . . . j ].

Hence f<}> = 0 mod [M^, Mg, . . . Mj](M/Mj^, M/Mg, . . .jM/Mj) = 0 mod M, and F = f(J)mod M = 0 mod M. Hence M = [Mj^, Mg, . . • jMj, (M, <{>)].

Now the spread of (M, (|)) is of the dimension n-r. Hence the same process can be applied to (M, ) as to M; and we finally arrive 152 at a module M,

Hence

^ ~ [Ql* Q2 » • * ■ * where Q^, Qg, . . . are all primary modules of ranks ^ r.

In more modern terminology, Lasker's theorem becomes

THEOREM XII-20, An ideal M in a polynomial ring R is the inter­ section of a finite number of primary ideals.

According to Zariskiand Samuel[222], Lasker’s theorem today is known as the Lasker-Noether Decomposition Theorem. This is due to the fact that E. Noether[163] (to be discussed in Chapter IV) discovered that Lasker’s theorem is a result of the "Ascending Chain Condition" on ideals. Thus this theorem stated in terras of Noether's contribu­ tions becomes

THEOREM III-21. In a ring R with the ACC on ideals, every ideal admits an irredundant representation as a finite intersection of primary ideals.

Remark

The ACC (ascending chain condition) on ideals and irredundant representations of idealsvwill be discussed in Chapter IV.

The proofs of both Hilbert's theorem and Lasker's theorem in their original forms are very involved and computational. They in­ volve terms which have become obsolete. At this point it becomes necessary to return to our more modern terminology; hence some explanations are in order. The following list reflects some of the name changes as well as changes in definitions which have occurred in modern ideal theory and hence mathematics in general: 153

1. The term "ideal" is used today as a replacement for the term

"module" of the past.

2. A "member of a module" is now called an "element of an ideal."

3. The "principal set of members" of a module is now called the

"set of generators" of an ideal.

. 4, "Elementary member" is now known as "generator."

5. "Simple module" is now a "simple ideal" (see Appendix A).

6 . A "member F of a module M is said to contain M" means that

"F is divisible by M" (see Appendix A).

7. A "module M contains a module N" has been replaced by an

"ideal M is divisible by an ideal N" (see Appendix A).

8 . The "least module of a set of modules" is now known as a

"common multiple of a set of ideals."

9. The "least contained module" has become the "least common multiple" of a set of ideals.

10. The "greatest module" of a set of modules has become a

"common divisor" of a set of ideals.

11. The "greatest contained module" today is called the "greatest common divisor" of a set of ideals.

12. "Primary module" has become "primary ideal."

13. "Prime module" has been replaced by the term "prime ideal

(see Appendix A).

14. "Prime module corresponding to a primary module" has become

"prime ideal corresponding to a primary ideal" (discussed in Chapter

IV). 154

15. The "product of modules" is called the "product of ideals" in modern terras (see Appendix A).

16. The "residual of a module M with respect to a module N" is now referred to as the "quotient of an ideal M by an ideal N." (See

Appendix A.)

17. The "spread of a module" is the "cardinality of ah ideal" as a set.

18. The "rank of a module" is now defined to be the "number of elements in a minimal generating set of an ideal."

19. An "unmixed module" is now called an "equidimensional ideal."

20. A "mixed module" today is expressed by the term "multi­ dimensional ideal."

6 . Some Fundamental Concepts of Algebraic Geometry

We pointed out in Section 1 of this chapter that Kronecker took a different route from the one chosen by Dedekind. Though their works, both based on some of the results of Kuiraner, were equivalent, Kronecker*s work was motivated by certain functions associated with algebraic geometry. Hilbert and Lasker expanded some of these aspects of algebraic geometry. We should like to cite some of the fundamental results of algebraic geometry and, in particular, to mention the work of Max

Noether. Though Appendix B contains some of the basics of algebraic geometry, it is necessary that some fundamental "groundwork" be given here in order to appreciate fully the remainder of this chapter.

In algebraic geometry an equation

Xg, . . . , x^) = 0 155 represents what is called an (n-1 ) dimensional algebraic surface in n-dimensional space with coordinates ^2’ ' ' ' Closely associated with this concept is the concept of an algebraic manifold now known as a variety.

DEFINITION III-49. A variety M in An(L) is the set of common zeros of a finite number of polynomials f^, fg, . . . jfj. over a ground field L, that is, the set of all solutions of the equations

flCxf, Xg, . . . ,x^) = 0

fgCxjL, Xg, . . . jXjj) = 0

ffCxi, Xg, . . .jX^) = 0 .

Remarks

1. If we form the ideal a = (f^y fg, . . . ,f^) from the poly­ nomials f]_, fg, . . • jfj., we see that the common zeros of f^, fg, . . . f^ are the common zeros of all polynomials of the form

f = 8 1 ^1 + 8 2 f2 + • • ' + 8 rfr: that is, all of the f^'s belong to a.

2. In view of Remark 1, M is called the variety of the ideal a.

Varieties play a very important role in the study of algebraic geometry. Though there are many theorems and definitions pertaining to this structure, we shall list only a few basic results which we state without proof.

1. In every non-empty set of varieties M there exists a minimal variety M*, that is, one which contains no other variety of the set. 156

2. The intersection M A N of two varieties M and N is a variety.

3. The union of two varieties M and N, M(jN, is a variety.

4. A variety which can be represented as the union of two (non­ empty) proper subvarieties is called composite or reducible. Hence a variety which is not reducible is said to be irreducible or indecom­ posable over the ground field k.

5. A variety M Is irreducible over a field k if and only if the associated ideal is prime; that is, if "fg contains M" implies that f or g contains M.

6 . Every variety M defined over k can be represented as the union of finitely many irreducible varieties over k.

"Dimension" is a concept used very frequently in algebraic geometry. We mention It here only to point-out some of its connec­ tions with ideals. In Section 4 of this chapter we defined "generic point." Van der Waerden[205] utilized this concept to define dimen­ sion and to prove some related theorems. Let (xj^, Xg, . . . ^ ) be a generic point over k of an irreducible variety M, in other words, a generic zero of the associated prime ideal P. Let r be the degree of transcendency of the system {Xj^, Xg, . . . ; then there are exactly r algebraically independent elements among the x^, say x^^, Xg, . . . yx^, the others being algebraically dependent on these. We may take the indeterminates tj^, tg, . , . jtj. for the x^, xg, . . . then all the

Xj^ are then algebraic functions of these r indeterminates. The degree of transcendency remains unchanged if the generic point is taken by a field isomorphism into another generic point (x^» xg, . . . ÿx^). Hence r depends only on P. 157

DEFINITION III-50. r is called the dimension of the prime ideal

P or of the variety M.

Remarks

1. The dimension of the prime ideal P f (1) may clearly be any number from 0 to n, where n is the number of elements in Xg, • • $■

V-

2. If (x^, Xg, . . .,x^) is a generic zero of a prime ideal P and(xj^, Xg, . . . ,x^) an arbitrary zero of the same ideal, then to each polynomial f(x^, Xg, . . . ^x^) of K[x^, Xg, . . . jX^] there corresponds the polynomial f(x^, x^, . . . of K[x^, Xg, . . .

Van der Waerden[205] proved

THEOREM III-22, If a prime ideal is zero-dimensional, all its zeros are generic and equivalent.

THEOREM III-23. A zero-dimensional irreducible variety consists of finitely many points which are conjugate over K.

THEOREM III-24. The distinct zeros of an r-diraensional prime ideal have transcendency degree = r;and if the transcendency degree is equal to r, the zero is generic.

THEOREM III-25. Every divisor P' of P has dimension r’ = r; if r* = r, then

P ’ = P.

DEFINITION III-51. The purely one-dimensional varieties

(varieties such that all of the components are one-dimensional) are called curves ; the two-dimensional ones are called surfaces; and the

(n-1 ) dimensional ones are called hypersurfaces. 158

The above results are just a few of the many results which Illustrate the use of ideal theory in algebraic geometry. No discussion of classical algebraic geometry would be complete without some discussion of the one time leader of the algebraic-geometric school in Germany, the late Max Noether.

Macaulay[143] summarizes the life and major contributions of this great German algebraic geometer. Max Noether was born in

Mannheim, Germany on September 24, 1844, and took his doctorate at

Heidelberg in 1868. Although Noether was not so well known out of his own country as some other mathematicians, he was deservedly held in the highest esteem by his countrymen. After the death of Clebsch, he became the recognized leader of the algebraic-geometric school in

Germany. He brought himself early to the front by his celebrated and fundamental "Satz aus der Theorie der algebraischen Functionen" which we shall discuss below.

Noether became a professor of mathematics at Erlangen, where he remained for nearly fifty years. During his tenure at Erlangen, he was a most prolific writer. His name will always be intimately associated with the respected German mathematical journal, Mathematlsche

Annalen. Scarcely a year passed from the appearance of the second volume in 1870 until the eighty-third in 1921 in which he did not make an important contribution to its pages. With the publication of the forty-second volume in 1893, he formally joined the editorial staff.

His daughter, Emmy Noether, made many contributions to ideal theory during the decade of the 1920's and will be discussed in Chapter V, 159

Noether[164] formulated the following theorem in his famous paper, cited above, which formed the basis for the "geometric trend" in the theory of algebraic functions.

THEOREM III-26. If for each zero a = {a^^, a2 , • • • of a zero-dimensional ideal M the exponent o is determined as the least natural number a for which

= 0 mod(M, holds with P = (xj^-aj^, X2"^2 » ' * • and if a polynomial f satisfies the condition

f = Pifi+P2 f2 for all zeros, then

f = 0 mod(M), where f^ and f2 are polynomials in two variables, and Pj^ and Pg are formal power series.

Though Max Noether’s works did not reflect an interest in the ideal aspect of algebraic geometry, one can safely conclude that he was Influenced by Kronecker. He influenced Hilbert and Lasker since some of his work became the models for their more algebraic theories.

On the other hand, Macaulay was a mathematician who was directly in­ fluenced by the works of Hilbert, Kronecker, Lasker and Noether.

7. Macaulay and His Modular Systems

Francis Macaulay (1862-1937) was an extraordinary English mathematician. In our development of Ideals, we observe that Macaulay is the first mathematician of non-German descent to make a significant contribution to this subject. Baker[11] summarizes his biography. 160 which includes a listing of his contributions to mathematical knowledge,

Macaulay, the son of a minister, was educated at Kingswood

School, Bath, and Saint John’s College, Cambridge. From the very beginning, he demonstrated an exceptional ability in mathematics.

Macaulay spent most of his life teaching in secondary schools.

For two years he was a master at his old school Kingswood, which served sons of the Methodist clergy. For the long period 1885-1911 he was a teacher of the mathematical scholarship boys (15 to 18) at Saint Paul's

School, London, where he prepared boys to enter a university. Two of his students there, G. Watson and J. Littlewood, later became eminent mathematicians. Many of his students were awarded scholarships to prestigious universities including thirty-four to Cambridge. In spite of his keen interest in the mathematical training of young men with its associated long hours of teaching duties, he found time for mathematical research.

He issued a volume on geometrical conics in 1895. From 1895 to

1905 his publications concerned algebraic plane curves, their multiple points and intersections; a plane curve C is defined to be C:f(x, y) =

0, where f is a complex irreducible polynomial. During this period, he was also greatly concerned with the theorem of Max Noether and the

Riemann-Roch Theorem (stated below).

THEOREM III-27 (Riemann-Roch). If the product aa' of two divisors is equal to the divisor !*f —^ of a product f , where is a d divisor of f(x, y), f(x, y) is a form, then the orders a, a' of a and 161 o' and the numbers y, y ' of linearly independent multiplies of and — , satisfy the relations (for a genus p. See Appendix C)

y = y'+a-p+l

a+a* = 2 p-2 ,

Gillispie[60] points out that it was this particular theorem and its

implications which led Macaulay to initiate his research on the theory of algebraic polynomials and polynomial ideals. (See Appendix C.)

Baker[11] observed that Macaulay seemed to have, ignored, somewhat, much of the existing foreign literature on algebraic geometry, the subject of the majority of his research. (This is particularly evident by his overlooking of the outstanding paper of Bertini of 1894 on algebraic geometry.) He, however, was attracted to the works of

Hilbert, Kronecker, Lasker and Noether, all of whose works have been cited previously. All of these great German mathematicians displayed an interest in algebraic geometry and its polynomial ideal theory, a

subject in which Macaulay made his most significant contribution to the

development of additive ideal theory.

Macaulay[142, 145] wrote two very important papers on polynomial

ideals. In 1913 he published a paper "On the Resolution of a Given

Modular System into Primary Systems including Some Properties of

Hilbert Numbers." In this paper Macaulay made his first attempt to

uncover some of the basic relations of polynomial ideals and the

associated concept of Hilbert numbers. The essence of a Hilbert number is as follows :

DEFINITION III-52. The dialytic equations of a polynomial ideal

are the equations obtained by equating all elements of the ideal to 162

P2 Pn zero and replacing every product Xg . . . by a corresponding unknown W , Thus if ^ 1 ,P2 ' • • • jPn

Pi P2 Pn F = Ea_ „ X, x„ . . . X Pj»P2 ». . . ,Pn 1 2 n is an element of the polynomial ideal, then

la W = 0 Pljp2» • • •JPnPl»P2* • ' 'fPn is a dialytic equation.

Remarks

1. Wp p is of degree P,+P,+ • • • +P„- 1 > 2 * * * * ♦ n

2. Each dialytic equation is of the same degree as the poly­ nomial or the element to which it corresponds.

3. The dialytic equations of an H-ideal are homogeneous.

DEFINITION III-53. The modular equations of an ideal are the equations which are identically satisfied by the coefficients of each and every member of the ideal.

DEFINITION III-54. The Hilbert number H(jL) of an H-ideal M is the number of independent modular equations of degree I.

Remarks

1. Hilbert[8 6 ] proved that when £ is large enough, H(£) becomes a polynomial function f (£) of £ of degree n-r-1, where r is of rank M.

2. The Hilbert number and function of a K-ideal are calculated from the corresponding equivalent H-ideal.

Macaulay’s paper of 1913 was followed by a more significant one in 1916 entitled "The Algebraic Theory of Modular Systems." In the first paper he explored the polynomial ideals (modules) that Hilbert 163 and Lasker had studied,their particular works having already been cited earlier in the chapter. The second paper is primarily an exten­ sion of the first and again one finds the works of Hilbert and Lasker.

However, this paper includes sections on unmixed ideals and inverse systems. The work on inverse systems and unmixed ideals was original.

Thus this particular work was Macaulay's unique contribution to ideal theory.

Macaulay[144] wrote a paper in 1934 which summarized the two earlier papers of 1913 and 1916. This paper involves a study of modern algebra and polynomial ideals. Since a modular system is now more specifically defined as a polynomial ideal, we shall cite excerpts from this paper along with excerpts from the other two, in order to discuss the basic properties of unmixed ideals, inverse and related concepts. We shall consider the unmixed (equidimensional) ideals first.

DEFINITION III-55. A polynomial ideal of rank r having a basis consisting r members only is called an ideal of the principal class.

(Note that the polynomial ideal (F^, F2 , . . . jF^) of rank r is of the principal class.)

DEFINITION III-56. Any ideal among Q^, p2 > • • • jQj^ which is contained in the L.C.M. of the rest is called reducible.

Remark

If

“ [Qj_> Q2 > • . • » then it is understood that all reducible ideals have been omitted.

(The remaining ideals are called Irreducible primary ideals.) 164

DEFINITION III-57. The irreducible order (cardinality) is de­ fined to be the order (cardinality) of an irreducible primary ideal.

DEFINITION III-58. An irreducible order which is not contained in another order of higher dimension is called an isolated order and the corresponding ideal is called an isolated primary ideal.

Remark

The other irreducible orders, those contained in orders of higher dimension, are called imbedded orders and the corresponding ideals, imbedded ideals.

DEFINITION III-59. An equi-dimensional ideal is an ideal whose irreducible orders, both isolated and imbedded, are of the same order.

DEFINITION IIÏ-60, A multi-dimensional ideal is one having irreducible orders of at least two different dimensions.

Macaulay[142] proved the next theorem, which was later applied to other ring-theoretic situations,

THEOREM III-28. A polynomial ideal of the principal class is equi-dimensional.

Proof:

Any polynomial ideal of rank n is equidimensional since it re­ solves into primary ideals which are all of rank n. Also, a poly­ nomial ideal of the principal class of rank 1 is equidimensional.

Hence the theorem is true for two variables since in this case the ideal can be of rank 1 dr 2. Hence we shall assume that the theorem is true for n-1 variables and prove it for n variables. We shall assume that the members of the basis have been modified, if necessary. 165 so that when (F^, F^, . , . ^F^) is of rank r, then (F2 , F3 , . • • jFj.) is of rank r-1 .

We prove first that an ideal

M = (F]^, ^2' ' ' ' 'V of rank r < n can not contain any irreducible simple ideal by showing that

(x^“Cn)F = 0 mod M requires that

F = 0 mod M no matter what value, special or otherwise, c^ may have.

Let

= X^P^+ X 2 F 2 + . . . +X^F^! then

( % + X2 F2 + . . . +Xrïr>x = c ° ° n n and

= C = 0 Fg, . . . ,F^)^ ^ ^ . n n n n

But (F^, Fg, . . . ,F^) is a polynomial ideal of rank r-1 in n-1 variables, so that by the assumption all of its irreducible orders are of rank r-1, and (F,) does not contain any of them. Hence ^ *n = <=n

. c ° 0 mod (Fj. F3 . . . . . , n n n " i.e., = X j^2^2'*'^13^3'*' ' ' ' '^^lr^r'^^^n~*^n^^l' Substituting this value for in the equation

(X2 F2 +X2 F2 + . . . +X^Fp^ . c = 0. n 166 we have 0 .

Hence by the same reasoning as before,

Xj+Xj^jFi = X23F3+ . . . +X2xF ^ +(^- c„)Ï2,

X 3 +X1 3 F2 +X2 3 F2 = % 3 ^F^+ . . . +X3xV

Xr+X2rFi+X2rF2+ • • . +Vl,rVl ' Multiplying these equations by P^, . . . and adding we have

XgF^+XgFgf . . . +X^F^ = ' ' ‘ +^r^r> » all the terms EX^jF^Fj(i< j), cancelling on both sides. This implies

that

F = Y]^F^+Y2F2+ • • * +'^r^r “ M) , and that (F^, Fg, . . . jFj.) does not contain any Irreducible simple ideal. (The are also polynomials Also we let .1 ^ be constants)

Now if (F^, Fg, . , . jFj.) were multidimensional, then for some value of s-r+2, the module (F-j^, Fg, . . . ,F^, Xg-a^, . . . jX^^-a^) would contain an irreducible simple ideal. But it does not, because it is of the principal class. Hence (Fg^, Fg, . , . is equidimensional,

Macaulay's theorem has been extended to rings other than the polynomial ones. Krull[l06] defined a local ring as follows:

DEFINITION III-61. A commutative ring R with unit element which has precisely one maximal ideal is called a local ring (Stellenring).

Remarks

1 , A local ring can also be defined as a commutative ring in which the non-units form an ideal. 167

2. A local ring R with a unique maximal ideal M can be trans­ formed into a topological ring by introducing powers of M as a system of neighborhoods of zero. (Topological rings will be discussed in

Chapter VI.)

DEFINITION III-62. Let R be a local ring of dimension d and let

M be its unique maximal ideal; then R is called a regular local ring

if M is generated by d elements.

Remark

A regular local ring can be defined as a local ring of dimension n generated by (u^, U2 , . . , u^) such that the ideals (u^, U2 , . , »•

Uj^) i = 1, 2, 3, . . j n are all prime ideals.

DEFINITION III-63. A local ring is said to be complete if every

Cauchy sequence (see Appendix C) has a limit in R.

DEFINITION III-64. An ideal a in a regular local ring R is equi-dimensional if all of its associated prime ideals are of the same

dimension. (Definition III-64 is a convenient alternatedefinition.)

Cohen[25] extended Macaulay's theorem (Theorem III-28) to regular

local rings and thus proved the next theorem.

THEOREM III-29. Let R be a regular local ring and

01 (a^: &2 * • * * be an ideal in R having a basis of r elements. If a is of rank r , then a is equidlmensional.

Zariski and Samuel[222] state and prove Macaulay's theorem in a more modern form.

THEOREM III-30. Let I be an ideal in

R = k[xj^, Xg, . . , x^] 168

of dimension n-k. If I is generated by h elements F 2 , . . ,

then I is equidimensional.

In view of the preceding developments, there exists a class of

rings called Macaulay rings. Zariski and Samuel[222] define a

Macaulay ring as follows:

DEFINITION III-65. Let R be a local ring. The common number of

elements of the maximal prime sequences in R is called the homological

codimension of R and is denoted by Cod h(R). If

Cod h(R) = dim(R),

we say that R is a Macaulay ring.

Remark

A Macaulay ring may also be defined as a ring in which

Macaulay’s theorem holds.

Macaulay[142] later developed a rather complicated modular

system called an inverse system. As the name implies, an inverse

system is an inverse structure for a modular system. His work on this

structure is very cumbersome in terras of very long proofs of theorems

and complicated systems of notations. In spite of the fact that this work was original, it does not play a significant role in the later

development of ideal theory. In view of this fact, we shall not give

a discussion of this work. Therefore, we leave Macaulay's works to

look at those of van der Waerden who is the next mathematician in the

sequence of researchers in additive ideal theory. 169

8 . Van der Waerden and His Reduction of Multi-dimensional Ideals

According to Turkevich and Turkevich[196], Bartel L. van der

Waerden was b o m on February 2, 1903 in Holland. He is the only mathematician who is associated with the early development of additive ideal theory thus far who lives in this decade (the decade of the

1970's). He maintains residency today in Switzerland, He is also the only mathematician thus far who is associated with two periods in the development of ideal theory. He greatly contributed to the development of the ideal theory which evolved from the polynomial ring theory following the works of Kronecker, Hilbert, Lasker, M. Noether and

Macaulay, In Chapter IV, we shall see that he is associated with the ideal theory of the decade of the 1920's, when the two theories of ideals appear to fuse.

After receiving his Ph.D. in 1926 from the University of Amsterdam, van der Waersen attended the lectures of Artin and E. Noether, Accord­ ing to Weyl[127], van der Waerden used notes from the lectures of both

Artin and Noether in order to write his famous Modern Algebra which is still in print today. Van der Waerden[205] included many concepts related to polynomial rings and algebraic geometry in this book. This was no surprise since he expressed an interest in algebraic geometry from the very beginning.

In 1927, van der Waerden[202, 209] published two papers in

Mathematische Annalen which were devoted to the polynomial ring theory of algebraic geometry. One of these papers focused on the reduction of multi-dimensional ideals to zero-dimensional ideals and utilized primary and prime ideals whose more modem definitions are 170

DEFINITION III-6 6 . An ideal q is said to be primary if

ab 5 0 (q)

and

a i 0 (q)

implies that there exists an e such that

b® = 0 (q).

DEFINITION III-67. An ideal p is said to be prime if

ab = 0 (p)

implies either

a = 0 (p) or

b = 0 (p).

Remarks

1. In the residue class ring modulo p, the definition of primary ideal may be stated in the following manner: if ab = 0 and a / 0, then b® = 0 for some e (see Appendix A).

2. In Section 5, a primary module (polynomial ideal) is defined

to be a module whose spread is irreducible, which is equivalent to the definition of primary ideal stated above (see Definition III-45).

3. A prime ideal p in a ring R may be defined to be an ideal such that its residue class ring ~ has no divisors of zero.

4. In Section 5, a prime module was equivalently defined such that if the product of any two modules contains it, then one or the other contains it (see Definition III-44).

Van der Waerden's[205] method of reduction can be summarized as follows: 171

If q Is a primary ideal in the polynomial ring K[x] (see Appendix A) of dimension d, p the prime ideal belonging to it (formally defined later in our discussion), {a^, a^, . . , a^} its generic zero, and a^, 3 2 » . . ^ a^ are algebraically independent (see Appendix A), then the ideals p and q are transformed into zero-dimensional Ideals by the substitution x^ = a^, Xg = Sg, . . , x^ = a^. This substitution can be carried out for all polynomials of the polynomial ideal q; thereby the polynomials q^ go over into polynomials q^ in K(a^, ag, . . , a^)

[x^_^^, x^^g, . . j x^] which generate an ideal q’. It is sufficient

then to perform the substitution x^ = a^^, ^ 2 " ^2 * ^ 3 " ^3 * ' ' ' x^ = Sj in the basic polynomials q^, qg, . . ^ q^; the corresponding polynomials q^, qg, . . , q^ generate the ideal q*, where

q' = (q{, qg, ' ' , q^-

The ideal q' consists of the polynomials divided by polynomials q^ in a^, Sg, . . , a^ which are distinct from zero; for the polynomials q^ form an ideal in K[a^, ag, . . , a^, x^^^, x^^^* • • 3 which

will generate an ideal in K(a^, ag, . . , ^*d+l* *d+2 ’ * * * ^n^ soon as the denominators g are admitted. An ideal p', generated from p, is generated in the same manner as q' is generated from q. In general, using the methods above, for every ideal

M = (f^, fg' ' ' 1 there is an ideal

M ’ = (f^, f', f» , . . , f^).

Geometrically speaking, the substitution x^^ = a^, Xg = ag, . . .

Xj = a^ Implies that all manifolds under consideration are cut by a 172 linear space = a^, x^ = a^, , . j x^, which passes through the generic point of the manifold of q.

If f(x^^, Xg, . . , x^) is a polynomial and f ^2’ * ' * ^d* *d+l’ * * * ^n^ belongs to q' and it can be shown by the above discussion that

' ièq . --à7?T - .-, a^) ' " ° therefore

g(a, x) = g(a)f(a, x).

Since a^^, a^, . . ^ a^ are algebraically independent, q(x) = g(x)f(x) = 0 (mod q).

Therefore, in order to determine whether a polynomial f(x) belongs to q, one needs only to investigate whether the corresponding polynomial

^(^1' ^2* ' ' ' ^d* ^d+1' ' ' ' *n^ belongs to q'. Van der Waerden[205] proved that q ’ uniquely determines q. In order to understand van der Waerden's proofs of certain theorems pertinent to the reduction theorems, several definitions are required.

Since van der Waerden was a student of E. Noether, it is no coincidence that he used some of the same basic definitions and concepts which she used. This will become more evident in the next chapter (Chapter IV) when we explore Noether's decomposition theorems. Van der Waerden[209] was concerned with extending some theorems for zero dimensional ideals to multi-dimensional ideals. This was his reason for reducing multi­ dimensional ideals to zero dimensional ideals.

DEFINITION III-68. Let q be a primary ideal and p be a prime ideal. Then the following relations are always valid: 173

q = O(mod p)

p*^ 5 OCmod q); the smallest number a for which these relations are valid is called the exponent of q.

DEFINITION III-69. An Irredundant representation of an ideal is a representation such that no component q^ includes the intersection of the remaining components.

DEFINITION III-70. Let

M = [q^, ^2» • • J be an irredundant representation such that all prime ideals p^ are distinct, where p^ belongs to q^ and the intersection of two or more primary ideals q^ cannot form a primary ideal. (This kind of repre­ sentation is calleda representation by greatest primary ideals and the greatest primary ideals are also calledthe primary components of M.)

DEFINITION III-71. A component ideal of an ideal a is any inter­ section primary ideals which appear in an irredundant representation of the ideal a by greatest primary ideals,

DEFINITION III-72. To every primary ideal q, there corresponds, a prime ideal p which may be defined as follows; p is the totality of elements b such that a power b*^ lies in q. p is said to be the prime ideal belonging to q . q a primary ideal belonging to p .

DEFINITION III-73. Let

ct = [q^, q2> • • > q&] be an irredundant representation in a ring R and 174

“l ^ *-^1’ Qg' ' ' » or 01^ = R, if k = 0

«2 = t^n+1’ • • ' q^^]. or = R if k = £.

Then

a = ot^ncxg,

and is a component ideal of a. The component ideal is said to be

isolated if no prime ideal p^^^ belonging to «g is divisible by a p^

belonging to a^.

DEFINITION III-74. A prime ideal belonging to a is imbedded if it

is a divisor of another prime ideal belonging to a.

Van der Waerden[205] proved the following theorems:

THEOREM III-31. The ideal q ’ in K(a^, a^, . . , ^*d+l* *d+2*

. , , x^] is primary; the prime ideal belonging to it is p ’; the

exponent of q' is equal to the exponent of q; the generic zero of p '

is { * ^d+2* ‘ * * and the dimension of p' is 0.

THEOREM III-32. If q is a primary component of an ideal

M = (f^, fg, . . , f^), then q' is also a primary component of the

corresponding ideal

M' = fj, , . , fj.).

THEOREM III-33. If q is an isolated component of M, then q' is

also an isolated component of M.

Van der Waerden's method for reducing a primary (multi-dimensional)

ideal to a zero-dimensional ideal provides the tools for determining

whether a particular polynomial f belongs to a given ideal

M = (fjL* fg. . . j f^).

Van der Waerden*s works utilize the ideal concept extensively.

Ideals in algebraic geometry are more or less a tool; van der Waerden's 175 works attest to this principle in spite of the fact he contributed to the development of ideal theory. We shall observe in the next chapter that van der Waerden's works also fall into the category of the ideal theory of our next period in the development of ideals. It was during this period that mathematicians seemed to have joined forces in their study of ideals. This, however, does not imply that there was a halt in the research activities of algebraic geometry. On the contrary, algebraic geometry began to develop at a rapid rate, van der Waerden contributing along with many other mathematicians. It just appeared during the next period that the paths of Dedekind and Kronecker officially met. Emmy Noether was the leader of this period, the decade of the 1920*s. CHAPTER IV

IDEAL THEORY DURING THE DECADE OF THE 1920'S

1. Emmy Noether and Her Decomposition Theorems

The development of ideal theory during the decade of the 1920*s was directed by perhaps one of the greatest mathematicians of this century, Emmy Noether. We summarize Weyi's[217] excellent biography of this great mathematician.

Amalie Emmy Noether (1882-1935) was born in the small south

German university town of Erlangen. She was the daughter of Max Noether whom we mentioned in the last chapter in connection with his outstand­ ing work in algebraic geometry, Emmy and her brother Fritz grew up in a home where the atmosphere was conducive to their early mathematical maturity.

In 1875 her father. Max Noether, was appointed professor of mathematics at Erlangen. He joined Professor Gordon, who had become a professor of mathematics at Erlangen in 1874, and became a very close friend of Gordon. It was this friendship which fostered an environ­ ment that nurtured the mathematical development of Emmy and Fritz.

Max Noether and his friend Gordon were constantly discussing mathe­ matics in the Noether home thus exposing the Noether children to the frontiers of mathematical research.

It was no surprise that Emmy Noether wrote her doctoral disser­ tation under Gordon, The title of her dissertation, written in 1907,

176 177

^was "On Complete Systems of Invariants for Ternary Biquadratic Forms."

(Her brother, Fritz, became interested in applied mathematics. After

receiving his doctorate, he became a professor at Technische Hochschule

in Breslau, Germany. He, like Emmy, was later driven out of Germany.

He became associated with the Research Institute for Mathematics and

Mechanics in Tomsk, Siberia.) Emmy, unfortunately in a sense, lived

during a time when women were just beginning to be accepted in

scientific careers. There were still many opponents of this new

policy; therefore, though women were accepted in universities as

students, they were not necessarily given positions in the universities

when they finished, the feeling being that most of the men scientists

and scholars still did not consider women to be the equals of men.

Noether's career was constantly marred by incidents of this type of

narrow-mindedness.

Already at Erlangen in 1913, Noether lectured occasionally,

substituting for her father when he was ill. During World War I in

1916, Emmy Noether went to Gottingen, where she was admired and re­

spected by both Hilbert and Klein. She was able to assist both of

these two great mathematicians because of her knowledge of the theory

of invariants. Hilbert and Klein were Involved with the general

theory of relativity and required a knowledge of certain aspects of

invariant theory. In particular, she was able to help these two

mathematicians with differential invariants and Euler's equations in­

volving an invariant multiple integral,

Noether was successful in reducing the problem of differential

invariants to a purely algebraic one whose solution was within their 178 reach. She accomplished this by the use of "normal coordinates."

Second, Euler had proposed certain identities that contained the con­ servation theorem of energy and momentum in the case of invariance with respect to arbitrary transformations of the four world coordinates

(x, y, z, t). These identities were between the left-hand sides of certain equations of variation which occurred when a certain multiple integral was invariant with respect to a group of transformations involving arbitrary functions. Neither Hilbert nor Klein was able to analyze this situation and propose an equivalent simpler problem which was within their reach. In view of this outstanding work, Hilbert contended that she deserved a position on the faculty of Gbttingen.

Also during the war, Hilbert tried to push through Noether's

"Habilitation" on the Philosophical Faculty in Gottingen. He failed due to the resistance of the philologists and historians. It is a well- known anecdote that Hilbert supported her application to the faculty at Gottingen by declaring at a faculty meeting, "I do not see that the sex of the candidate is an argument against her admission as

'Privatdozent.' After all, we are a university, not a bathing estab­ lishment." He probably provoked his adversaries even more by that remark. Nevertheless, she did give lectures at Gottingen which were announced under Hilbert's name. In 1919, after the war and the proclamation of the German Republic, certain conditions changed.

Women were allowed to play a more important role in academic life.

Hence Noether's habilitation became possible.

In 1922, she was nominated a "nichtbeamteter ausserordentlicher professor." This was a mere title carrying no obligations and no 179 salary. This position is somewhat comparable to a postdoctoral fellow in an American university. (Graduate students in American universities do receive some financial remuneration.) (One observes from the declension of the adjectives in nichtbeameter ausserordentlicher professor that even with this small title Noether was referred to as a man. Many professors and colleagues did this out of their so-called

"respect" for her as a mathematician. They still thought of a woman as being a creature inferior to a man. Many of them even referred to her as Herr Noether instead of Fraulein Noether.) Finally Emmy was entrusted with a "Lehrauftrag" for algebra which carried modest remuneration. In these modest positions she worked at Gottingen until

1933. During the last few years of her tenure at Gottingen, however,

she held these lowly positions in the beautiful new Mathematical

Institute which had risen in Gottingen due to the energy of Courant and the generous financial help of the Rockefeller Foundation. By this

time, she was without doubt the strongest source of mathematical activity there considering both the fertility of her scientific research program and her influence upon a large circle of students.

Noether lived in close communion with her students; she loved

them and took an interest in their personal affairs. They formed a

rather noisy and stormy family; the "Noether boys" as they were called

at GottingeiC' Among her students were Hermann Krull, Holzer, Grell,

Koethe, Deuring, Fitting, Witt, Tsen, Shoda and Levitzki. All of these

students later made profound contributions in the field of modern

algebra. Schmidt, a later significant mathematical figure, was greatly

influenced by Noether through Krull, who taught and greatly influenced 180 him. Van der Waerden (discussed in Chapter III) came to her from

Holland as a mature mathematician with his own ideas, but he learned

from Noether the apparatus of notions and the kind of thinking that

permitted him to formulate his ideas and to solve his problems. In

fact, van der Waerden[205] in the second volume of his classic Modern

Algebra utilized much of her work. This was also true for Deuring in

the case of his book on algebras. Noether herself had been influenced

by the works of Dedekind, whose style was quite different from her

teacher Gordon, who was a computational type of mathematician.

Of her predecessors in algebra and number theory, Dedekind was

the one mathematician whose works were closely related to Noether’s

philosophy of mathematics. She felt a deep veneration for him to the

extent that she expected her students to read Dedekind’s appendices to

a book by Dirichlet[42] in all of its editions. She took a most active

part in the, editing of Dedekind's works; here the attempt was made to

indicate, after each of Dedekind's papers, the modern development

built upon his investigations. She lived through a great flowering of

algebra in Germany, toward which she contributed much.

Noether's scientific production seems to fall into three distinct

epochs: (1) the period of relative dependence, 1907-1919; (2) the

investigations grouped around the general theory of ideals, 1920-1926;

and (3) the study of the non-commutative algebras, their representa­

tions by linear transformations and their applications to the study of

commutative number fields and their arithmetics, from 1927 on. It is

in the period 1920-1926 that we focus our attention. 181

Noether developed a general theory of ideals on an axiomatic basis. Her chief axiom was the "Teilerkettensatz," which; means . divisor chain condition:

If a chain of ideals Og, . . . in a ring R is given and if every is a proper divisor of that is, c a^_j_2^,then the chain breaks off after a finite number of terms. This axiom became a part of the hypothesis of many of her investigations. In other words, Noether primarily worked with rings in which this axiom was valid and in an axiomatic manner she deduced other results. Her "Teilerkettensatz" is sometimes called the maximum condition and takes on several equivalent forms as we shall observe later in this section. By her abstract theory she was able to deduce how many important developments in mathematics are "welded" together. Her analysis of Lasker’s theorem

(proved in Chapter III) is a good example of her axiomatic powers.

With her "Teilerkettensatz" in the hypothesis, she was able to give a much simpler proof of Lasker’s theorem. (We shall assume that the rings of this section satisfy Noether's condition.)

Noether[163] gave an account of her "Teilerkettensatz" and all of the main results known on the existence and partial uniqueness of normal decompositions. Her "Teilerkettensatz" is also known as the

"finite condition for ideals" and the "divisor chain condition for ideals." Since she was the first mathematician to deduce such profound results by assuming-that the rings involved satisfied her axiom, these particular rings are now called Noetherian rings. The work of Emmy

Noether[163] on decomposition was motivated by the following: 182

In the ring of integers, every integer a > 0 is the product of distinct prime numbers ®1 ®r a = Pi ?2 •*•?!.» and analogously every ideal (a) is the product of powers of prime ideals :

(a) = (Pi)^^(P2 )^^ • • •

However, in more general rings, the decomposition of ideals is not this simple. In general an ideal will not have a representation as a product; at most an ideal will have a representation as the L.C.M, of components where the L.C.M. of an ideal M = [«i, «2 , . , .] is

«1(102^ * • • • This representation is, however, similar to the representation of (a) above; that is,

(a) = Kpi^^), (pg^^), • • Î CP^ ^)]*

The ideals Cpi^^) have the characteristic property; if p^®^[ab but p^®i I a, then p^^ [b; that is, the factor b must contain at least a factor of p^ . This implies that b® must be divisible by p^^^i for some power e. Therefore,

ab = 0(pi®^), and

a i CPi^^)

b^ = O(pi^i).

Ideals with this property are called primary ideals (discussed in

Chapter III).

Although Hilbert proved his Basis Theorem for a polynomial ring over a field, it can be extended (see Chapter III) to rings in which 183 every ideal has a finite basis. This condition can be shown to be equivalent to the following equivalent statements:

Divisor Chain Condition, First Statement: If a chain of ideals

c%2 , Og, • ■ . in R is given and if every is a proper divisor of a^:

® i ‘^“i+l’ the chain breaks off after a finite number of terras.

Divisor Chain Condition, Second Statement: If an infinite chain

of divisors a^, «2 » ®3* • * • given:

“i*= “i+l> all terms must be equal after a certain n:

“n = % + l “ • • • • Divisor Chain Condition, Third Statement (Maximal Condition): If the divisor chain condition is valid in R, then in every non-empty set of ideals there is a maximal ideal; that is, an ideal which is con­ tained in no other ideal of the set.

Divisor Chain Condition, Fourth Statement (Principle of Divisor

Induction): If the divisor chain condition is valid in R and a prop­ erty E is valid for every ideal a (in particular for the unit ideal) as soon as it is satisfied by all proper divisors of a, then property

E is valid for all ideals.

Suppose that R is a ring in which the divisor chain condition is valid. The following theorem limits the values of the exponents of

Ideals :

THEOREM IV-1. A power of p, p®, is divisible by q, where p is a prime ideal belonging to the primary ideal q; that is, 184

p® = 0(q).

Proof :

Let (p^, Pg, . . r Py.) be a basis for p and let p^^^, pg ; e p^ lie in q. If we set

r e = 2 (e,-l)+l, i=l then p® is generated by all products of the p^, e at a time. In each such product, at least one factor p^. must occur more than (e^-1) times, and therefore at least e^ times. All generators of p® are in q. Hence p® is divisible by q.

Remark

The exponent e is an upper bound of the exponents of the powers to which the elements of p must be raised in order to obtain elements of q.

In the theorems to follow, the divisor chain condition is valid in the rings.

THEOREM IV-2. If ag = 0(q) and a i. 0(q), then there is a power a such that 8° 5 0(q).

Proof

It is sufficient to choose o = e. If aS = 0(q) and a ^ 0(q), it is true, from above, that 3 = 0(p) and hence

3^ = 0(p®) = 0(q).

Now that we have pointed out the concepts pertinent to the de­ composition of integers into their prime factors and some of the analo­ gies between numbers and ideals, we want to demonstrate Noether’s decomposition of ideals and the associated concepts. 185

DEFINITION iy-1. An ideal M is said to be reducible if it can be represented as the intersection of two proper divisors (ideals):

M = a n 8, aoM, goM.

(If such a representation is not possible, the ideal is said to be irreducible.)

Remark

The primeideals are examples of irreducible ideals. (If a prime ideal p had a representation, p = a

ag = O(ang) 5 0(p), a 0 0(p), g i 0(p) which would contradict the assumption that the ideal is prime.)

THEOREM IV-3. Every ideal is the intersection of a finite number of irreducible ideals.

Proof;

Let M be an ideal. If M is irreducible, then we are done.

Hence let M be reducible; that is,

M = o n g, aZ) M and go M.

If we assume that the theorem is valid for all proper divisors of M, then it is valid in particular for a and g. Hence let us say that

a = [i]^, ^2» * * >

^ " f^s+1’ ^s+2» * ' Î ^r^* This means that

M — [ij^, ^2 * • • • ig, ^s+1* • * > > hence the theorem is also valid for M. Since the theorem is valid for the unit ideal, which is always irreducible, it is valid in general by the "principle of divisor induction." 186

The fact that every ideal is the intersection of a finite number of irreducible ideals gives rise to several theorems whose proofs are given by Noether[163]. We list these theorems below.

THEOREM IV-4. Every irreducible ideal is primary,

THEOREM IV-5. Every ideal is representable as the intersection of a finite number of primary ideals.

THEOREM IV-6. The intersection of a finite number of primary ideals which belong to the same prime ideal is again primary and be­ longs to the same prime ideal.

THEOREM IV-7, An irredundant intersection of a finite number of primary ideals, which do not all belong to the same prime ideal, is not primary.

THEOREM IV-8. Every ideal has an irredundant representation as the intersection of a finite number of greatest primary components.

(These primary components belong to distinct prime ideals.)

Thus far in our discussion of the decomposition of ideals into primary components and irredundant components, we have not specified whether or not these compositions or representations are unique.

Noether[163] established some conditions,

THEOREM IV-9. In two Irredundant representations of an ideal M by greatest primary components, the number of components is the same and the prime ideals belonging to these components are also the same.

(The components themselves need not be the same.) 187

Remarks

1. This theorem gives rise to a set of ideals Pg» . • » Pg which are uniquely determined and are the prime ideals belonging to an irredundant representation

a = [qj^, 92» ' " ^ qg]'

These ideals are called the prime ideals belonging to an ideal a.

2. The properties of the prime ideals belonging to an ideal a, which are most significant are

a. If an ideal a is divisible by one of the prime ideals

belonging to an ideal g, then

g:a = g

and conversely.

b. Let a be the principal ideal (a). If an element a is

not divisible by a prime ideal belonging to an ideal g, then

g:a = g;

that is

ac = 0(g)

implies that

c = 0(g).

c. Let a = [q*, q^, . . , q ’], where the q^ are primary

ideals. If no prime ideal belonging to a is divisible by a

prime ideal belonging to g, then

g:a = g

and conversely. 188

Noether[163] considered isolated components to prove an impor­ tant theorem. We shall state and prove this theorem after we establish a pertinent definition.

DEFINITION IV-2, An isolated component ideal of an ideal ot. can be defined to be one such that the set of prime ideals belong­ ing to it either contains no imbedded prime ideals or contains, for every imbedded ideal, at least all the prime ideals in which it is imbedded.

Remark

Though some of van der Waerden's results were discussed in the preceding chapter (Section 9), we can definitely see the influence of his teacher, Emmy Noether.

THEOREM IV-10. Every isolated component ideal of an ideal a is uniquely determined by giving the prime ideals belonging to it.

(Note that this theorem implies that every isolated component ideal of an ideal a is uniquely determined by a subset of all prime ideals belonging to a.)

Proof;

Let two representations of a be given by

a = where has the same prime ideals belonging to it as a^. Now we let

and be isolated components. By property 2a of prime ideals belonging to an ideal cx (above), we have

“l-“2 ” “1

" “ r 189

But since a = “ ct^na^, we have, by forming quotients mod 02' that

implies that

“l - *1'

Similarly^

implies that

“i - “l*

Hence

“ 1 = “ r

Therefore is uniquely determined.

Corollary. The isolated primary components of an ideal are uniquely determined.

The preceding theorems are only the beginning of the imaginative powers of Noether. Noether, due to her power to use the axiomatic approach, was the first mathematician to use a ring as an abstract mathematical system rather than limiting herself to the rings of in­ tegers of algebraic number fields, the rational field,and the field of rational functions.

Noether[163],in her work on the existence and uniqueness of decomposition, immediately caught the attention of most of the im­ portant algebraists of that era and was accepted as a major contributor to the field of mathematics. As was pointed out above,

Noether was greatly involved in both the personal and research affairs of her students. In view of this practice of hers, her students' 190 works reflected her Influence. Krull, one of her students at

Gottingen, wrote a paper which was based on her paper "Ideal theorie in Ringbereichen."

2. Krull and His Application of Ideal Quotients to A Fundamental Theorem of General Ideal Theory

Wolfgang Krull, according to Turkevich and Turkevich[146], was born on August 26, 1899. He was born near the small city of Baden-

Baden, Germany. He was one of Emmy Noether's most successful students, having studied with her at Gottingen during the period 1919-1921. He, however, earned his doctorate from the University of Freiburg in 1921.

A very prolific mathematical researcher, he frequently published in

Journal fur die reine und angewandte Mathematik, Mathematische

Annalen, Mathematische Zeitschrift and other scholarly scientific journals of Europe.

In 1923, he published a paper in Mathematische Annalen which

"grew-out" of the paper of Noether[163] cited above in Section 1.

Noether stated and proved that each ideal can be represented as the least common multiple of a finite number of primary ideals belonging to different prime ideals in (Noetherian) rings. Krull[107] demon­ strated how this theorem can be applied by use of the concept of an ideal quotient. He also applies the "uniqueness" theorem via the ideal quotient and defines some of the fundamental concepts of ideal theory in a new way by using the ideal quotient. In order to prove

Noether's theorem cited above, Krull utilized the concept of a primary ideal and its relevant prime ideal. The following lemmas are 191 pertinent to this development and are summarized and proved by

Krull[107].

LEMMA IV-1. Each ideal a can be represented as the least common multiple of a finite number of primary Ideals belonging to different prime ideals.

Remark

The proof of this lemma is done in three stages;

a. Each non-primary ideal is reducible; that is, it can be represented as the least common multiple of the real divisors. (See

Noether[163] paragraph number 4, Theorem 6 .)

b. Each ideal can be represented as the least common multiple of a finite number of primary ideals (see Noether[163] paragraph number 2, Theorem 2).

c. If q^, q2 > q^, • • j q^^ are primary ideals, then [q^, q2 >

. . j qj^] is primary if and only if all the q^ belong to the same primary ideal p, and in this case [q^, q2 > . . j 1 belong to p

(see Noether[163] paragraph number 5, Theorem 8 ).

The concept of an ideal quotient is defined as follows:

DEFINITION IV-3. By the ideal quotient 0 :8 , where o is an ideal of a ring R, we mean the totality of all elements r of R such that

rb = 0 (o) for all beg.

The following lemma, in five parts, can be used to perform certain calculations with ideal quotients:

LEMMA IV-2. Let o, o^, o^, . . , o^ and 8 » 8 ^^, 6 2 he ideals in a ring R, then the following conditions hold: 192

a. If $2 - O(S-j^), then a:6^= OCacGg).

b. (a:gi) : 6 2 = a;(@2 ^2 ^ “

c. [o^, ttg, . . , a%]:g = [a^^G, «2 :8 , . . j

d. ^1^2 [&!, $2 ^ relatively prime to a if and only if and Gg are relatively prime to a.

e. If G2 is relatively prime to a, then a:[G^, G2 ] =

LEMMA IV-3. If q is a primary ideal with the exponent a belong­ ing to p and if a is an arbitrary ideal non-divisible by q, then q:a is primary and belongs to p. q:a is different from q; that is, q:a is relatively prime to q, if and only if a is divisible by p.

The preceding lemmas make it possible to prove a fundamental theorem about uniqueness.

THEOREM IV-11. If

a = [qii 92» • • J = [q{, 92» • • » 9^] are two shortest representations of a by greatest primary ideals; that is, no component q^, and respectively q^, can be left out; and if p^ (respectively p^) means the prime ideal belonging to q^ (re­ spectively q^), then I ~ Z' and with a suitable numeration p^ and pj^

(i = Ij 2, . . , Æ.) are identical.

Proof;

By the first lemma (Remark c), the theorem holds for & = 1.

Hence the theorem must be proved for & >1, First, we suppose that for Z a p^ can be chosen in such a way that there exists no real divisor of p^^ among the prime ideals p^. Further, we can suppose, without loss of generality, that none of the prime ideals p|, pg,

. . , p% ^ is identical to p^. We can then form by Lemma IV-3 193

0:9% = [qi:9A, ^ 2 ' % ^ * ' ’ Sz'S^' ' '

= [QI, 92' ' ' * 4%-l] “ [Si' ' • » 9&,_1, qjiîqj,]. Now since the theorem is valid for & = 1, since [q^, 92» • • j is obviously a shortest representation of a:q^, and since &' = &, then certainly Z' > JL-1 and one of the ideals q^, q^, . . , q \ q^,:q^ is a real divisor ofq^, and hence by Lemma IV-3

«:9j^ = [qi. 92' ' ' ' 9^_i] *

Hence by our assumption, stated above,

pj : ocp;,)-

Now since p^ cannot be a real divisor of p^^, then

and hence [q^, q^, . . , 9^t_i3 as well as [q^, qg, . . , q^_^l is a shortest representation of a:q^. Hence

&'-! = 5,-1 so that 5,' = Z and thus

Pi “ Pi (i = 1, 2, . . , &-1).

Hence our theorem is proved.

It is therefore possible to demonstrate that p^ can be chosen in such a way that there occurs no real divisor of p^ among the ideals p!, p', . . f p*, . The demonstration or argument is as follows: If 1 2 & such a choice is not possible, then p^f could be determined in such a way that none of the ideals p^, p^, • . , P^'_i» Pi' P2’ ' ' : P% would be a (real or unreal) divisor of p^, Consequently we would obtain: 194

a:q%, = 92 = 41'' * * ’ = [^^iq*,. 92:9%'' * * » li'’ " (si' si' - ' ' si'-il = Isi. Sj. • • . q^l - “ and [qj^, q^, . . , q^i] under our assumption above, would not be a

shortest representation of a. We can, therefore, always choose p as

indicated and thus the theorem is valid,

Krull's[107] paper more or less parallels the, paper of Noether

[163], We further support this statement by stating the other theorems which he proves in the paper. One can observe the corresponding

theorems in Noether's paper.

THEOREM IV-12. a is relatively prime to G , where a and G are

distinct ideals, if none of the prime ideals belonging to a is divis­

ible by one of those belonging to G.

THEOREM IV-13. A component ideal of a is clearly defined by

its relevant prime ideal if it satisfies the following condition: If

p^ is an arbitrary prime ideal belonging to p an arbitrary prime

ideal belonging to a, and if

p = O(p^),

then p also belongs to

Remark

in the sense of the theorem, is called an isolated component

ideal of a.

THEOREM IV-14. If

« “ [9l> 92» • • > 9%] and a = [q[, q^, . . , q|] are two shortest representations of a by greatest primary Ideals and

if q^ and q| belong to the same prime ideal pj^, then 195

u = [q^i 92* * * ’ 9^_2» 9j^* ' * > 9%] " 92» • • j 9^_2»

91» 9j_+^» . . 3 9%].

THEOREM IV-15. Each ideal can be represented as the least com­ mon multiple of the smallest mutually relatively prime ideals.

THEOREM IV-16. If a and 6 are two arbitrary ideals, then

a:6 = a:B^

if and only if a;B = is an isolated component ideal of a or is

identical to (1) , If is an arbitrary isolated component ideal of a without being identical to a, then there always exists an ideal 3 which satisfies the equation

a:G - a^.

THEOREM IV-17. If a is an arbitrary ideal, p is a prime ideal

and if

a:p° = a:p*^^ = f a;

that is, if « 2 is the unit ideal or an isolated component ideal in the

sense of the preceding theorem, then p belongs to ot if and only if

there exists no real divisor p' of p which satisfies the equation

a:p‘ = a^.

We shall cite another of Krull's major contributions later on in

this chapter. We return to the "master-teacher," Emmy Noether, to

point out another of her contributions to ideal theory. This work, on

the isomorphism theorems, will be followed again by a work of one of

her students, who was motivated by this particular work of hers. 196

3. Eimy Noether and Her Isomorphism Theorems

Emmy Noether's creative powers were in her uncanny axiomatic approach via her keen imagination. In her paper of 1927, which appeared in Mathematische Annalen. Emmy Noether[162] proved some theorems con­ cerning rings which may be applied to other basic algebraic structures such as groups, modules, vector spaces, etc. The summary of these re­ sults as applied to rings are based on the definitions of Appendix A.

THEOREM IV-18 (First Isomorphism Theorem). Let R and S be rings and let f;R^S be a ring homomorphism from R onto S. If a = ker f, then

- = S. a Proof:

Clearly a is an ideal of R. For let a.hsa. Observe that

f(a-b) = f(a)-f(b) = 0-0 = 0 which implies that a-bect. Observe also that

f(a'b) = f(a)'f(b) = O' 0 = 0.

Therefore, a is a subring of R. Now let aea and reR. Observe that

f(ra) = f(r)f(a) = f(r) 0 = 0.

Therefore, raea. Similarly,

f(ar) = f(a)f(r) = 0f(r) = 0. D Therefore, area. Thus a is an ideal of R. Hence — is a quotient ring. a Define f : — + S by a f (r-fn) = f (r). f is well-defined since

r+d = s+a implies that 197

(r-s)+a = a which implies that

r-sea = ker f so that

f(r-s) = f(r)-f(s) = 0

f(r) = f (s).

Also observe that

f [ (r+a)+(s+a) ] = £[(r+s)-fa] = f(r+s) = f(r)+f(s) = f (r+a)+f (s+«).

That is,

f [(r+ot)+(s+a) ] = f (r+a)+f (s-fo) .

Similarly,

f [ (r+a) (s+ct) ] = f(rs+ot) = f(rs) = f(r)f(s) = f (r+a)f (s-fo) ,

Therefore f is a ring homomorphism. Clearly f is onto since f is onto,

Finally, suppose

f (r-Kt) = f(s+a) .

Then

f(r) = f(s)

which implies that

f(r)-f(s) = 0

which implies that

f(r-s) = 0

which implies that

r-sE ker f = e

so that

(r-s)-fa = ot. 198

Thus r+a = s+a.

Hence f is one-to-one. Therefore

S. a THEOREM IV-19 (Second Isomorphism Theorem). Let a and g be sub­ rings of a ring R with a an ideal of R. Then angis an ideal of g, a+g is a subring of R and

ÇL+Ê. - . 3- • a a ri g Proof ;

We must show that a+g is a subring of R, Let a^+b^,0 2 + 6 2 ea+g, where a^^, a2 ca and b^, b^eg . Observe that

(ai+bi)-(3 2 +6 2 ) “ ^1 ‘*‘^ 1 ’'^ 2 ~ ^ 2 “ (a]^-a2 )+(b]_-b2 )ca+g since a and g are subrings of R. Similarly,

(a-j^+b]^) (32+62) ° a]_a2+aj^b2+bj^a2+b^b2 “ (&i&2'*'&1^2'*'t'l&2)'*^^l*'2EO'*'B since a is an ideal of R and g is a subring of R. Therefore a+g is a subring of R, Also a is an ideal of R. Now define f : by f(b) = b+a. Now observe that

bi = b; implies that

bi+ u» 6 2 + a so that

fih{) = £(62) hence f is well defined. Also observe that

£(6 2^+6 2 ) = (62^+6 2 )+» = (6 2^+0 )+(6 2 +0 ) = f (b2,)+f (6 2 ) and

£(6 ^6 2 ) = 6 2^6 2 + 0 = (b^+a) (bg+a) = f(b2 ^)f(b2 ). 199

Therefore f is a ring homomorphism. Now observe that

ker f ={beg|b+o = a} = {beB|bea} = g O a = oHg .

But in the proof of the first isomorphism theorem, she showed that if f:R-^ S is a ring homomorphism, then ker f is an ideal of R. Hence a r) g is an ideal of g. Then by the first isomorphism theorem

a H^ g = Im f. But since ct is an ideal of R, then a+g = g+a. Hence

= { (b+a)+a j aea, beg} = {b+a|beg}. a Hence Im f = —^ . Therefore

a+g _ g a a rig THEOREM IV-20 (Third Isomorphism Theorem). Let a - g - R and g R let a and g be ideals of the ring R. Then - is an ideal of — and

R a „ R . 1 g Proof : a

Define f ; ^ ^ by f (r+a) = r+g. If r+a = s+a, then

r+a = s+a implies that

(r-s)4a = a so that

r-sea - g hence

(r-s)+g = g;

that is

r+g = s+g. 200

Hence f is well defined. Also observe that

f[(r+a)+(s+a)] = f[(r+s)+a] = (r+s)+g « (r+g)+(s+6) = f(r)+f(s).

Also

f [(r+a) (s+a) ] = f(rs+a) = rs+a = (r+a) (s+a) = f(r)f(s).

Therefore f is a ring homomorphism.

Clearly Im f = , Hence by the first isomorphism theorem

R _« ____= R . ker f 3

But

ker f = {r+e|1 ER and r+G = G) = {r+ajreR and reG) = ^ .

Therefore

R a „ R

È ' 3 * a These theorems demonstrate the power of the quotient rings via ideals. Since an ideal is an additive abelian group, one can observe how these theorems can be applied to groups, modules, vector speaces, etc.

Gray[68] points out the fact that an Immediate consequence of the first isomorphism theorem is the following corollary:

Corollary. A ring homomorphism is injective (see Appendix A) if and only if its kernel is the zero ideal.

Proof: =**»>

Suppose ker f = (0); that is, ker f is the zero ideal, where f:R + S is a ring homomorphism. Let a, be ker f. Hence

f(a-b) = f(a)-f(b) = 0-0 = 0 , that is, a-bekerf. But 201

ker £ = 0 implies that

a-b = 0.

Thus

a = b.

Therefore, f is one-to-one.

Proof ; < =

Suppose f is one-to-one and let f(a) = O', where O' is the zero of S. Now clearly

f(0) = O'.

Hence

f(a) = f(0).

But f is one-to-one which implies that a = 0. Therefore ker f = (0).

The third isomorphism theorem is utilized in proving the follow­ ing theorem which is sometimes called the correspondence theorem.

(Gray[68] refers to this theorem as the "Lattice Theorem.")

THEOREM IV-21. Let R be a ring and let a be an ideal of R; then there exists a one-to-one correspondence between the set S of subrings of g and the set cT of subrings of R containing a.

Noether's work, just cited above, was the beginning of her interest in non-comrautative algebras, their representations by linear transformations and their applications to the study of commutative number fields and their arithmetics. Grcll, a student of Noether, utilized some of her results in his developments of the extension and contraction of ideals. 202

4. Grell and His Contractions and Extensions of Ideals

Friedrich August Heinrich Grell was born on February 3, 1903 in

LUdenscheid, Germany according to Foggendorff[171]. He was one of the so-called "Noether-boys" of Gottingen due to his studies with Emmy

Noether, whose students formed a very close family on the campus. As one would expect, due to the great influence that Emmy Noether had on her students, Grell's works reflect the Noether approach.

Emmy Noether's paper of 1927, along with a few others, enabled

Grell to study the contraction and extension of ideals. Grell[69] in his paper of 1927 made a study of "fields of modules and ideals." We require, however, some preliminary definitions. We look initially at modules.

DEFINITION IV-4. Let M be a system of elements A, B, . . , .

In M, let there be an "equality relation" satisfying the three rules of reflexivity, symmetry, and transitivity, so that any two elements

A, B of M are either equal (A= B) or unequal (A B) . Furthermore, let there be in M four operations such that for every ordered pair A, B of

M, a sum A+B, a difference A-B, a product AB, and a quotient A;B are defined. Then M is called a system of quadruple composition.

DEFINITION IV-5. Two systems and Mg are called isomorphic if there exists a one-to-one correspondence between the elements of M^ and Mg such that sums, differences, products and quotients are preserved.

DEFINITION IV-6 . A system M is called a modular field and its elements A, B, C, . . . are called modules if the four operations, +,

-, X, :, satisfy the axioms (note that the module defined here is not 203 equivalent to the polynomial ideals defined in Chapter III) :

I.

a. A+B = B+A.

b. (A+B)+C = A+(B-fC).

c. A+(A-B) = A.

II.

a. A—B — B“A.

b . (A-B)-C = A— (B-C) .

c. A-(A+B) = A.

III.

a. AB = BA.

b. (AB)G = A(BC).

Remark

An isomorphism between two module fields is called an arithmetic isomorphism to distinguish it from ordinary field isomorphisms.

The three sets of axioms cited above were used by Grell to establish the theorems below:

t h e o r e m IV-22. A+A = A.

t h e o r e m IV-23. A-A = A.

The axioms in set I and set II are connected by

1 . First Distributive Law: If A+C = C, then A+(B-C) = (A+B)-C.

2. Second Distributive Law; A(B+C) = AB+AC.

3. Third Distributive Law; (A-B):C = A :C-B:C.

The operations of addition and subtraction are sufficient to define a modular ring as a partially ordered set. A partially ordered set is defined in accordance with Hausdorff[76], 204

DEFINITION IV-7. A partially ordered set is a set in which for any two elements A and B, it is always true that exactly one of the following notions holds: A = B (A is equal to B) or A

B), or A > B (A follows B), or A [|B (A is incomparable with B). In - particular, in M

A = B for A+B = A A-B = A

A < B for A+B = A A-B f A

A > B for A+B A A-B = A

A I I B for A+B 9^ A A-B ^ A .

DEFINITION IV-8 . A module A is called a multiple of B if A SB, and is called a divisor of B if A — B.

Remarks

1. In the case of a multiple, it is evident that A-B = A, and in the case of a divisor, A+B = A.

2. If we omit the equality sign, then A is respectively a proper multiple or a proper divisor of B.

DEFINITION IV-9. A module D is called the greatest common divisor of the modules A and B if

1. D is a common divisor of A and B, and

2. Every common divisor of A and B is a divisor of D,

DEFINITION IV-10, A module V is called the least common multiple of A and B if

1. V is both a multiple of A and B, and

2. Every common multiple of A and B is a multiple of V, 205

Remarks

1. The greatest common divisor of two modules A and B is A+B.

2. The least common multiple of A and B is A-B.

In a somewhat analogous manner, ideal fields may be defined and their related concepts are required for our later developments in terms of contracted and extended ideals.

DEFINITION IV-11. A modular field is a field of ideals or an

"ideal field" if it contains a module Q with the following properties;

1 . Q is a divisor of every module of the modular field.

2. For each module A of the modular f ield,AQ ZA.

Then the elements of the modular field are called "ideals" of the ideal- field I.

Remarks

1. In ideal fields one essential fact is that the product of elements is divisible by each factor, that is, AB - A and AB^b ; for we know that B - Q and hence AB-AQ-A, and likewise for AB-B.

2. If an ideal field contains an ideal e such that ae = a for every ideal a in I, then the ideals, 6 , Y» • * • are called relatively prime if their greatest common divisor is e.

DEFINITION IV-12. An ideal p is called a prime ideal of I, if a < II p and B <[|p together imply ag < || p.

DEFINITION IV-13. An ideal q is called a primary ideal of I if (for every natural integer n) ot ^||q and <||q imply that aG < || q.

We are now in a position to state Grell's defintions for con­ tracted and extended ideals. Let E be a ring and P an arbitrarily chosen subring. 206

DEFINITION IV-14. The extended Ideal M in I belonging to an ideal m in P is defined to be the least common multiple of all ideals of E which are divisors of m as P-modules.

Remark

For every ideal m of P its extended ideal M exists because E it­ self is an ideal and certainly is a modular divisor of m.

DEFINITION IV-15. The contracted ideal n in P that belongs to an ideal N in E is defined to be the greatest common divisor of all the

ideals of P which are multiples of N as P-modules.

Remarks

1. For every ideal N of E its contracted ideal n exists be­ cause the null ideal of P is a multiple of N,

2. The contracted ideals in P and the extended ideals in E are all called "distinguished" ideals.

3. The ideal M is said to be generated by ra and the ideal n is generated by N.

The structure of the class of distinguished ideals is as follows:

The contracted ideal n generated by N is the least common multiple

N-P of the P-modules N and P. For each n must be a multiple of N and

P and hence of N-P. As for the remaining, N-P is an ideal in P and is divisible by N. N-Pin, so that n = N-P. Thus n is the set-theoretic intersection^NnPj of N and P.

The extended ideal M generated by m is the greatest common divisor m+raE of the P-modules m and mE. Hence M must be a divisor of m and mE, and therefore of M4ME. Furthermore, because

(m+raE)E = mE+mE^ - m+mE, 207

the module m+mE must be an ideal in E and a divisor of m, M i m+raE,

therefore m+raE = M. M consists of the totality of elements

■■u+yiai+ . . . +UrOr, where p, Pg, . . , run through the elements of m and where

^2 , • • Î through all elements of E. If E contains a unit

element e, we have that m i mE and then

M = m+mE = mE,

Some additional facts or properties of the distinguished ideals with regard to the operations, which Grell[69] proves, are stated below as theorems.

THEOREM IV-24. The greatest common divisor

D = M^HMgf . . . M^+ . , .

of the extended ideals Mj^, M 2 , • . , M^., . . . i s itself an extended

ideal and can be generated by

5 = • • • •

THEOREM IV-25. The greatest common multiple

V = nj^-Ug— • • • —n^— . . . of contracted ideals n^^, ng, . . . n^ with generated ideals , Mg,

. . a Nj., . ... is itself a contracted ideal and can be generated by

V = -Mg- ......

THEOREM IV-26, The product of two extended ideals M^ and Mg with

generating ideals m^ and mg is again an extended ideal and is generated by m^'mg.

DEFINITION IV-16. Let Q be a primary ideal with radical P; 0 is

said to be strong primary if there is an m such that P™cQ. 208

THEOREM IV-27. Let the Ideal Q in E be a strong (or weak) primary ideal with F as its associated prime ideal. Then in P, the contracted

ideal Q-P = q, is a strong (or weak) primary ideal with the contracted

ideal P-P = p as its associated prime ideal.

Remark

The (finite or infinite) exponent of q is not greater than the exponent of Q (see Noether[162]).

Though Grell’s[69] paper gives a thorough discussion of contracted and extended ideals, he consistently expresses these concepts in terms of a class of ideals; that is, an ideal field. These concepts, perhaps, will be more meaningful to the reader if they are expressed in terms of two

ideals and in more modern equivalent terminology. Zariski and Samuel[222]

summarize these concepts in terms of the conditions we desire.

DEFINITION IV-17. Let R and S be two rings with identities. Also

let f be a homomorphism of R into S such that f(1) =1 . If A is an ideal c —I in S, then the ideal A = f (A) is called the contracted ideal or the

contraction of A.

DEFINITION IV-18. If a is an ideal in R, the ideal a® = Sf(a)

generated by f(a) in S is called the extended ideal or the extention

Remarks

1. When R is a subring of S, the ideal U is the Intersection

RHU, it contains every ideal of R which is contained in U and is thus

the largest ideal in R contained in U.

2. The ideal is generated by a in S; it is contained in every

ideal of S which contains a and is thus the smallest ideal in S which

contains a. 209

3. a® consists of all elements of S of the form s^a^+Sgag^

. . . +s^a^, where n is an arbitrary positive integer, s^sS, a^ea, i — 1, 2,3, • • » n«

Several significant relations exist among the distinguished ideals of R and S. Namely

1. If AcBjthen A^cSC; if ac.6 , then a®ccB®.

2 . a '^^c t A} a ^ c ^ o t .

3. = AC. ct^ce ,= a' _e

4. (A+B)c^ A C + b C* (a+G)® = a®+G®.

5. (A+B)® = ACq b C; (aOG)® a® n G®-

6 . (AB)®oA^B®; (aG)® = a^G®*

7. (A:B)®<=A®;B®; (a:G)® = a®;G®.

Remarks

1. In view of relation 3, we can say that if an ideal in S is an extended ideal, it is the extension of its contraction, and that if an ideal in R is a contracted ideal, it is the contraction of its extension.

2. If we denote by (E) the set of all extended Ideals in S, and by (G) the set of all contracted ideals in R, the mappings A-»-A® and a a® are 1-1 and are inverse mappings of (E) onto (C) and of (C) onto

(E).

3. The 1-1 correspondence between the ideals in (C) and (E) is an isomorphism with respect to the fundamental ideal-theoretic opera­ tions (sum, product. Intersection, and quotient) to the extent to which these operations do not lead to ideals outside of (C) or (E). 210

4. Relations 4, 5, and 6 show that the set (E) is closed under addition and multiplication and that the set (C) is closed under intersection.

THEOREM IV-28. The set (C) of all contracted ideals in R is closed under quotient formation.

Proof:

Let a and B be contracted ideals in R. Set A = and B = 3 ®, hence a = A® and 3 = B®. We must show that a:8e(C). This is achieved if we show that if A is any ideal in the ring S and Be(E) the set of extended ideals in S, then A®:B® = (A:B)®. With the aid of relation 7, it is only necessary to prove that

A®:B®c (A:B)®.

Observe that (A®:B®) B = (A®:B®)®B®®. But Be (E). Hence (A®:B®)®B®® =

(A®:B®)®cA®®cA. Hence (A®:B®)®trA;B and (A®:B®)®cA:B and (A®:B®)cr

(A®:B®)®®0 (A:B)®.

Grell[6 8 ] also Introduced the concept of a ring of quotients in his paper. This was done via a multiplicative system which we shall define below. This concept is significant for two particular reasons.

First, Grell observed that certain relations exist between the ideals of a ring R and its ring of quotients R^. Secondly, this concept is utilized to show that any integral domain may be embedded in a field.

McCoy[153] summarizes Grell's work in m o d e m terminology as is

Indicated below.

DEFINITION IV-19. Let p, q be arbitrary elements of M Now

if p, qeM implies that pqeM, then M is called a multiplicative system. 211

Let R be a commutative ring and let M be a set of elements of R which are not divisors of zero in R. Now we consider elements of the form (a, p) where aeR and peM.

DEFINITION IV-20. The pair (a, p) is said to be equivalent to

(b, q) if aq = bp. (Symbolically this is expressed by (a, p) 'v(b, q).)

Remark

The elements (a, p) may be distributed into equivalence classes with respect to the defined equivalence relation, (The class to which

(a, p) belongs will be denoted by [a, p].)

Suppose Rjjj is the set of all equivalenceclasses [a, p]. We shall make S into a ring by certain definitions of addition and multiplication in R . m DEFINITION IV-21. [a, p]+[b, q] = [aq+bq, pq].

Remark

[a, p] = [b, q] if and only if aq = bp.

DEFINITION IV-22. [a, p][b, q] = [ab, pq].

Remark

Since p, q are elements of the multiplicative system M, it is clear that pq is in M.

R^ is a ring. We also observe that if p and q are elements of M and a is any element of R, then

[ap, p] = [aq, ’q] because

paq - paq.

Also if 212

[ap, p] = [bp, p] then

ap^ = bp^ and since p is not a divisor of zero (since peM) now we see that

a •<-> [ap, p], where p is any element of M, defines a one-to-one correspondence between elements of R and a set R of elements of R . It is obvious that R_, m^ m T^l is independent of the choice of p. Also we can observe that

a+b [(a+b)p, p] = [(a+b)p^, p^] = [ap, p]+[bp, p] and

ab ^ [abp, p], where 2 2 [abp, p] = [abp , p ] = [ap, p][bp, p].

Hence we see that R^^^ and R are isomorphic.

Now we let T be the ring isomorphic to R^ in which the elements of R^^ are replaced by the corresponding elements of R. Hence in T, we shall write a instead of [ap, p]. The unit element in T is [p, p], where peM, In T, the equation

p[x, y] = [p, p] has the unique solution

[x, y] = [p, p^].

Thus p has an inverse in T. The elements of T are precisely all quotients ~, where aeR, peM. We have established:

THEOREM IV-29. Let R be a commutative ring and M a multiplicative system of elements of R which contains no divisor of zero. Then the set 213

T of all , that is, [a, p], where aeR, peM is a ring which contains

R as a subring, and in T all elements of M have inverses.

If R is an integral domain and M is the set of -all non-zero elements of R, then we have the corollary;

Corollary; An integral domain R may be imbedded in a field of quotients, that is, a field whose elements are of the form ^ , where a and b are elements of R and b ^ 0.

Grell[69] showed that certain relations exist between the ideals of R and those in R^. These relations involve the contracted and extended ideals which we developed earlier in this section. Again we shall appeal to Zariski and Samuel[222] who summarized Grell's work in modern terminology.

It was observed that the quotient ring R^ exhibited the following very interesting property:

There exists a homomorphism f of R into R^ such that:

1. The kernel of f is the set of all elements x in R for which there exists an raeM such that

xm =s 0.

2. The elements of f(M) are units in R^^. f(v') 3. Every element of R^ may be written as a quotient — . , where f (m) xeR, meM.

Two additional definitions are required before we state the results of Grell,

DEFINITION IV-23. An element x of a ring R is said to be prime to an ideal a of R if a:(x) = a. 214

DEFINITION IV-24. A subset E of R is said to be prime to a if each one of its elements is prime to a.

Remark

IVhen a is a finite intersection of primary Ideals, a subset E of

R is prime to a if and only if it is disjoint from the union of the associated prime ideals of a.

The relations between the ideals in R and the ideals in R^ are spelled out in the theorem which follows:

THEOREM IV-30. Let M be multiplicative system in a ring R with identity and let R^ be the ring of quotients with respect to M.

1. If a is an ideal in R, then a®® consists of all elements a in R such that amea for some meM. .

2. An ideal a in R is a contracted ideal; that is, a = a®®, if and only if M is prime to a.

3. Every ideal in R^^ is an extended ideal.

Proof (1):

Note that bea®® implies that f(b)ect®, where f is the homomorphism from R to Rm given above. But every element of R^ can be written as a quotient , where xeR, meM. Thus any element of a® may be written in f (m)

where x^^eR, m^eM, a^^eu. Now since M is a multiplicative system, it is closed under multiplication. Hence reduction to a common denominator yields m = 'Jm^^eM since any element of o® may be written in the form

where aea, raeM. Thus we see that bea^c and only if there f (m) exist aea and meM such that f(bra-a) = 0. But ker f is the set of all 215 elements xeR for which there exists meM such that xm = 0. Hence f(bm-a) = 0 implies that there exist an element aea and m, m'eM such that (bm-a)m’ = 0. Thus there exists m" = mm'eM such that bm"ea. Con­ versely, the element ra"eM gives f(b)f(m")ef(a) so that f(b)ea® since f(m") is a unit in that is, bea®®. Thus a®® consists of all elements beR such that bmea for some meM.

Proof (2) : = >

Suppose a is a contracted ideal of R. Then a = a®®. But by (1) a®® consists of all elements beR such that bmea for some meM. Hence a consists of all elements beR such that bmea for all meM; that is, each meM is such that a:(m) = a. Hence M is prime to a.

Proof (2) : < = =

Suppose M is prime to a. Then a:(m) = a for every meM and bmea for all beR. Thus a = a®® and a is a contracted ideal of R.

Proof (3):

Suppose a' is an ideal in R^; then any x'ea' may be written in the form

v' = ftx) fCm) ’ where xeR and raeM. Thus we have that f(x)ea', xea'® and x'ea*®® so that a* c a'®®. But by relation (2) a'®® tsa'. Hence a' = a'®®. Thus every ideal in R^^ is an extended ideal.

Grell's "distinguished" ideals and ring of quotients have many applications in mathematics, especially the ring of quotients. This concept will be utilized later with reference to other mathematical matters pertaining to ideals. Van der Waerden, whose works we have 216 already explored In connection with the Ideals evolving from the poly­ nomial rings of algebraic geometry, discovered an application of a multiplicatively closed set to some of the concepts already explored by E. Noether[162].

5. Van der Waerden and His Components of An Ideal Determined by a Multiplicatively Closed Set

Emmy Noether’s[163] paper of 1921, cited at the beginning of this chapter, was the most significant contribution to the mathematical world during the decade of the 1920*s. Not only was this true because of results on decompositions, their existence and uniqueness, but be­ cause of its effect on other mathematicians in decades to follow. Van der Waerden, whom we discussed in connection with the additive ideal theory of polynomial rings, was constantly applying ideal theory to geometrical problems. This fact was most evident in van der Waerden's

[203] paper of 1928.

In order to find the solution of classes of a system of n homo­ geneous equations

fl = 0, f^ = 0, . . , f^ = 0 of corresponding degrees m^^, m^, . . , m^ with n+1 unknowns Xq , Xj^,

. . , x^, one forms the "u-resultant" (discussed in Chapter III) of the system of equations; that is, the resultant of the forms f^, f^, • . , f^ of a general linear form Zuj^Xj^. The u-resultant, as a function of the u.jhas the degree " m. and resolves, in a suitable extension field, 1=1 into linear factors In general, this means that if the 217 coefficients of the forms are Indeterminates, these linear factors are all different. For special values, however, multiple factors can occur and it can happen that the u-resultants disappear identically. The latter statement means that there could exist an indefinite number of solution classes. Disregarding the latter statement, the coefficients

of each linear factor represent a solution of classes. A y-fold linear factor results in a "solution of multiplic­ ity M." The theorem of Bezout states that "The sum of the multiplicities of the solution of classes is hence " m.." In view of this theorem, i=l i we can see that the u-resultant furnishes the means of defining the multiplicity. The number of solutions in the "general" case is equal to the sum of the multiplicaters in the special case. This fact is sometimes called the "maintenance of the number."

In more geometrical terras, Bezout’s theorem means that the sum of the multiplicities is the point of intersection of n algebraic hyper­ surfaces in projective n-space. If the number of intersection points is finite, the sum of the multiplicities is equal to the product of

the degree numbers. In this case, the multiplicities are defined, as above, by the exponents of the linear factors of the u-resultant.

Also in more geometrical terms, Bezout's theorem can be expressed more comprehensively by stating "The sum of the multiplicities of an r- dimensional or an (n-r) dimensional manifold in projective n-space is equal to the product of the degree numbers of this manifold if the number of the points of intersection remains finite. In this case, of course, the degree of a manifold of complex dimension k is under­ stood to be the number of intersection points in general linear 218 n-space. However, the multiplicity of a point of intersection has not really been defined at this point in our discussion.

Van der Waerden[203] considers the problem for arbitrary values at n and r. He considers the following as his "general case." The two given manifolds for which points of intersection are unknown are brought into a general position, relative to each other, by linearly transforming the one with indeterminate transformation parameters and then bringing the manifold to the intersection. From this more general intersection problem, we can obtain the original one by specializing the transformation matrix U as a unit matrix E. Due to the definition of multiplicity and the maintenance of degree numbers. Van der Waerden's problem amounted to proving; If M and M ’ are algebraic manifolds of dimension r and n-r in projective n-space, and if one of them, say M, is subjected to a linear transformation U with inde terminate coefficients, then the number of points of intersection of M' together with the trans­ formed manifold MU is equal to the product of the degree numbers of M and M'.

Van der Waerden[203] proved the Bezout theorem by utilizing ideal theory as an auxiliary medium. He treated, in the paper, a process which produces ideals from ideals. In fact, this process can be used to produce the isolated components of an ideal. He used this process for examining the behavior of a polynomial ideal at a given point. In so doing, however, he uncovered a method determining the isolated com­ ponents of ideals by utilizing the concept of the component of an ideal determined by a multiplicatively closed set. Emmy Noether[163] proved that isolated components of ideals are uniquely determined by their 219

corresponding prime ideals (This was discussed in Section 1 of this

chapter.) Northcott[169] gives the modern summary of van der Waerden’s

findings which we cite below:

Somewhat like Grell[69], van der Waerden used a set M defined as

follows :

DEFINITION IV-25. Let M be a non-empty subset of a ring R such

that Va,beM, then a*beM.

Remark

This set differs from Grell's multiplicative system in that Grell

imposed the additional restriction that the elements be non-divisors of

zero.

DEFINITION IV-26. Suppose a is an ideal of a ring R and M a multiplicatively closed set ; we denote by the set of all elements

X such that cxca for at least one ceM. a is an ideal called the m isolated component of a determined by M . (a^ is sometimes simply called

the M-component of a.)

We require two lemmas in order to demonstrate van der Waerden's

result:

LEMMA IV-4. Let a be an ideal and let p^, p^, . . , p^ be prime

ideals, none of which contains a; then there exists an element aeo

such that no p^ contains a.

Proof:

The proof is accomplished by induction on the number of prime

ideals. If n = 1, the assertion is trivial since one could choose aea-p^. Now suppose we assume that Lemma IV-4 is true for n-1 prime

ideals; then for each i, 1 - i 5 n, there exists an element a^sn which 220

Is not contained in any of p^, p^, . . , p^_^, ^l+l’ * * * ^n' at least one i we have a^ ^ p^ there will be nothing further to prove; we can therefore confine our attention to the case in which a^ p^ for all i. Set n a “ J / V 2 • • • “i-i^i+i ■ - - then the jth terra in the sum does not belong to p^, but if i f j then a, . . . a. ,a.. a ep, since a, occurs in the product. Hence 1 i-1 i+1 n j a^p. for any value of j. Therefore aea since a.,a_, . . ^ a sa. Hence ] ± z n we have found an a which does not belong to any of the p^.

l e m m a IV-5. Suppose that

a = • • • ^1%* where the are p^-primary. Let M be a multiplicative set and let

p^HM ^ for m+1 ^ i £ n but not for 1 5 i < m. then

“m " • • • " ‘’in- Proof:

Let XGO^; then cxea = . . , Hq^, where c is a suitable element of M. Consequently, if 1 ^ i 1 m we have cxeq^ and c^p^, which

shows that xeq^^ for 1 - i - m. Now let ytq^A q^ A . . . Aq^. For each j > m we choose c^ep^ AM, and then for a sufficiently large N,

*'"Vl"V2'^ • • • y(c^j^c^2 ■ • • gut M is a multi­ plicative set, hence (c^,c,^2 * * * c^)^S .=> yeu^. Therefore

“m ' i^nq^n . . . flq^.

Corollary. A decomposable ideal has at most a finite number of

isolated components. 221

THEOREM IV-31. Suppose that an ideal a has a primary decomposi­ tion, and let a = . . . A q ^ be a normal decomposition of a, where q^ is p^-primary. If Plj^i Pig» • • j Pi^ isolated set of prime ideals belonging to a, then depends only on pi^, Pi^, . . . p^^, and not on the particular decomposition con­ sidered .

Proof :

Let M consist of all elements not contained by any of the Pi^,

Pi2 » Pig* * • i Pij." Hence M is multiplicatively closed. We shall show that

“m " li^nq^^A . . . Aqi^ from which the theorem itself will follow since is completely deter­ mined by a and the prime ideals Pij^, p^g, . . j Pi^. By Lemma IV-5, it is only necessary to show that every p^ which is not one of the pi^,

Pig, . . 1 Pij. intersects M. But since p^^, p^g, . . , p^^ is an isolated set for such a p^, we shall have that

Pj ^ Pij^> Pj ^ Pi2* " ' * Pj ^ Pi^'

Hence, by Lemma IV-4, p^ contains an element not in any of the pi^.

Pig, • • » Pij.* In other words,

P^Am ^ 4>.

Corollary. Let p be a minimal prime ideal of a; then the primary component corresponding to p is the same for all normal decompositions of a. 222

6. Krull and The Intersection Theorem

The concept of a multiplicatively closed set proved to be a

powerful tool for mathematicians working with ideals. Krull[113] in

his paper of 1928 used this concept to prove a very significant theorem

of ideal theory known as the "Intersection Theorem."

In order to demonstrate this theorem, however, we shall make use

of a known theorem which we stated and proved in an equivalent form in

connection with Emmy Noether's work in Section 1 of this chapter

(Theorem IV-1). We shall state this equivalent form for the sake of

convenience :

THEOREM IV-32, If R is a Noetherian ring and if q is a p-priraary

ideal, then p*^ £ q for some positive integer a.

In addition to the above theorem, we require the following lemma:

LEMMA IV-6. Suppcpse that a and S are two Ideals in a Noetherian ring R, and suppose that 8 S ng. Then there exists an element aea such

that

(l-a)g = (0).

Proof :

Since R is Noetherian, 8 is finitely generated, say

3 = bg, . . , b^).

Then b^eag, hence each

\ = ^il^l+^12^2'*' * * • +*10^0' where the a^j belong to «, consequently

r=l 223 where 6^^ = 1 if i = r, and = 0 if i ^ r. Utilizing the standard notation for determinants, if

A = Is..-a, 'ij~“ij'’ then

Abr = 0 for each r, and therefore

Ag = (0).

But A = 1-a, where aea. Hence there is an element aea such that

(l-a)g = (0).

THEOREM IV-33 (Krull's Intersection Theorem). Let a be an ideal oo of a Noetherian ring R; then an element x of R belongs to 0 a if and i=l CO 1 only if we have x = ax for at least one element aea. Further, H a is i=l an isolated component of the zero ideal (0).

Proof : = = >

Let M consist of all those elements which can be written in the form 1-a where aea. Observe that if 1-a and 1-b are arbitrary elements of M, then

(1-a)(1-b) = 1-b-a+ab = l-(a+b-ab).

But (a+b-ab)ea. Hence l-(a+b-ab)cM. Therefore M is multiplicatively closed. Further if we set

8 then the first part of the theorem asserts that 8 is the M-component

(0)^ of the zero ideal. Hence the second part of the theorem follows immediately from the first. 224

Proof ; < =

If now X = ax, where aea, that is, if xe(0)^, then 2 3 x=ax=ax=ax=. . . which shows that x belongs to every power of a. Hence (0)^ 5 g. In order to prove the opposite inclusion, we show first that 6 £ ag. Since

R is Noetherian, we may write

ag = q^AqgA . . . Aq^, where q^ is p^-primary. Thus for each i ag - q^, and therefore either g - or a E p^. But if a S p^ by Theorem IV-32, mentioned above, we can find an integer n such that p^ - then g £ a^ £ p” « q^, so that in any case g S q^. Since, for each i, g is contained in q^. we have g E ag. But by the lemma^g S (0)^. Hence g = (0)^^ Therefore the theorem is proved.

Krull[108] wrote a textbook on ideals in 1935 which became a classic. Thin text was published by Julius Springer in Berlin and later in the United States by the Chelsea Publishing Company in 1948. The bibliography of this textbook not only lists the many works on ideals by Krull himself, but it lists every work on ideals [238] prior to

1935. Among those listed are Holzer[89] and Hermann[82] who were also students of Emmy Noether. These mathematians also made contributions to ideal theory during the decade of the 1920's.

7. Artin and The Minimal Condition

Emil Artin (1898-1962) was perhaps the most important contributor to the development of ideal theory during the decade of the 1920's who was not a former student of Emmy Noether. However, there is an analogy 225 between their works on ideals in that Noether was concerned with rings which satisfied the maximal or ascending chain condition on ideals while Artin concerned himself with rings which satisfied the minimal or descending chain condition on ideals. Lang and Tate[131] edited a book which includes all of the major works of Artin. The preface of this book outlines a brief biography of Artin which includes the following.

Artin was born on March 3, 1898 in Vienna, Austria. After the death of his father, his mother remarried and moved to Reichenberg,

Bohemia. It was there that Artin obtained his "Reifepriifung" in 1916.

Among the universities where Artin studied are the University of Vienna,

the University of Leipzig, where he studied with Herglotz and received

his Ph.D. in 1921, and the University of Gottingen. He went to the Uni­ versity of Hamburg in 1922 and became "Privatdozent" there in 1923,

"Ausserordentlicher". professor in 1925 and "Ordentlicher" professor in

1926. Van der Waerden was fortunate in that he had the opportunity to

study with Artin as well as Noether. Artin*s interest in the minimal

condition was motivated by Wedderburn's work on hypercomplex number

systems.

In 1926 Artin became interested in hypercomplex number systems or

algebras as they were later named. Just as rings with the ascending

chain condition generalize the properties of polynomial rings, rings

with the descending chain condition generalize finite dimensional

algebras. Rings satisfying the descending chain condition are called

rings with the minimal condition.

Since an algebra (hypercomplex number system) is a ring, Artin

[6] concentrated on the ring aspect of this struture along with both 226

Noether's maximal condition (ACC) and the minimal condition (DCC). He later discovered that he did not really need the ACC in order to prove several results and therefore uncovered many significant facts about rings satisfying the DCC, We shall cite van der Waerden's[205] sum­ mary of these facts.

THEOREM IV-34. If R is a ring satisfying the minimal condition for left ideals and if a is an element of R which is not a right zero divisor in R, then the equation

xa = b can be solved in R for every b.

Proof:

In the set of left ideals Ra", u = 1, 2, . , . , there must be a minimal left ideal, say Ra^. Since

S Ea" is valid, but

Ra"^^cRa™ is excluded by definition, then

Ra"*^ = Ra".

Hence every product ba™ may be written in the form ca"^^^:

ba” - ca'^\

Since the factor a may be cancelled m times on the right and left, this equation may be reduced to

b = ca.

Hence the equation xa = b has a solution x = c. 227

Remark

If R is a ring satisfying the minimal condition for right ideals and if a is not a left zero divisor, then ax = b can be solved.

Artin[7] also worked extensively with nilpotent ideals. Thus we have

THEOREM IV-35. The sum of two nilpotent left ideals is a nilpotent left ideal.

Proof ;

Let “ ^2 ~ ' Then ^ is a left ideal; it is the totality of all sums whose summands are products of n+ra-1 factors from or In each summand, there must be at least n factors from

or m factors from S,^- H we assume that there are n factors from

then each term has the form

...... A 2^ ... J where the dots represent factors from and at least n of the factors are from 9.^. Therefore, since R2^ - then it must be true

...... 2 A* = (0), hence ^ = (0) ,

THEOREM IV-36. Every nilpotent left ideal (or right ideal) is contained in a nilpotent two-sided ideal.

Proof I

Let A be a nilpotent left ideal, that is, A*^ = (0). Then AR is also nilpotent since

(AR)° = A(RA)*"^R - AA^"^R ^ = A°R = (0).

The right ideal (A, AR) generated by A is accordingly the sum of two 228 nilpotent left ideals and therefore is a nilpotent left ideal. Hence it is a nilpotent two-sided ideal.

DEFINITION IV-27. If w is an element of a ring R such that w generates a nilpotent two-sided ideal, then w is called a root element.

THEOREM IV-37. All elements of a nilpotent left or right ideal are root elements.

Proof ;

If w is a root element contained in a nilpotent left ideal A, then by Theorem IV-36 above, w is also contained in a two-sided nil- potent ideal. Hence, the two-sided ideal generated by w is also nilpotent.

DEFINITION IV-28. The totality of root elements of a ring R is called the radical of R. (Radicals will be discussed in more detail in Chapter V.)

Artin[7] made a thorough investigation of rings without radicals

(see Appendix A). We shall state and prove several lemmas and a most significant theorem which was proved by Artin.

LEMMA IV-7. If A is a left ideal, a an element of R, then A is operator-homomorphic (see Appendix A) to Aa by the correspondence

X 4^ xa.

Proof ;

Observe that

(x+y)a = xa+ya

rx*a = r*(xa)

(Ax)*a = A*(xa). 229

LEMMA IV-8. A minimal (simple) left ideal £ is mapped by an operator homomorphism either onto the null ideal or the homomorhism is an isomorphism.

Proof:

K = {ae^l&a = O) is a left ideal and K - A. Hence K = (0) or

K = A, In the first case K is mapped onto A by 9 and 0 is an isomor­ phism.

LEMMA IV-9. A minimal left ideal A is either nilpotent (and then 2 we have that A = (0))or A contains an idempotent element e and is generated by this element:

e^ = e in A, A = Re.

Proof ;

Suppose we assume that

f (0).

Then there exists atA such that

Aa 3^ (0).

Then by Lemma IV-7, the correspondence x xa is a homomorphism and by

Lemma IV-8, this correspondence is an isomorphism which carries A into

Aa and

Aa + (0), Aa ^ A.

Hence

Aa = A.

Therefore every element of A may be represented in the form xa, where xeA, and in particular,

a = ea, 2 where esA, Hence e 0 and ea = e a. Therefore, the elements e and 230 2 e are mapped onto the same element by the isomorphism x->xa; therefore, they are equal to each other, that is, 2 e = e. 2 The ideal Re is not the null ideal since e = esRe and it is contained in A, therefore

Re = A.

Hence A contains an idempotent element e and is generated by e.

DEFINITION IV-29. If a (one-sided or two-sided) ideal a in a ring

R is the direct sum of one-sided or two-sided ideals respectively, say,

“ “ " ' * ®^n* where n > 1 and every 3® (0 ), then the ideal a is said to be (one­

sided or two sided) directly decomposable.

LEMMA IV-10. If e is idempotent and A = Re, then R is the

direct sum of A and another left ideal A'; that is,

R = A@A'.

Further, for all xeA

xe = X,

for all x 'e A'

x'e - 0 .

Proof:

Every a in R may be written as a = ae+(a-ae). The elements ae

form the left ideal Re = A. The elements a-ae also form a left ideal

A', since

(a-ae)-(b-be) = (a-b)-(a-b)e,

r(a-ae) = ra-(ra)e,

A(a-ae) = Aa-(Aa)e. 231

The elements a-ae are annihilated by e; that is, 2 (a-ae)e = ae-ae = 0, while the elements ae reproduce themselves when multiplied by e; hence 2 (ae)e = ae = ae.

The single element which is annihilated and reproduced when multiplied by e is the null element. Hence A and A' have only the null element

in common, i.e., the sum R = A#A ' is direct.

THEOREM IV-38. A ring without radical satisfying the minimal

condition for left ideals is the direct sum of simple left ideals.

Proof :

The theorem is valid for the null ring. Suppose we assume,

therefore, that R is distinct from the null ring and that A^ is a minimal left ideal distinct from the null ideal. By Lemma IV-9 and

Lemma IV-10, we have that

R = A^BA'.

If A' is not the empty ideal, then there must be a minimal left ideal

Ag such that Ag - A' and A g f (0). Then again by Lemma IV-9 and Lemma IV-10

R = A g ® A * .

But if we apply this sum representation of the elements of R to the

elements of A', we obtain

A’ = Ag+A".

Hence

R = Aj^+Ag+A".

Again if A " ^ (0), we seek in A " a minimal left ideal A ^ f (0) and

find as above that

R = A^+Ag+A^+A'",. 232 etc. Hence the series R 3 A' 3 A" o A'" => . . . must contain a minimal left ideal by the minimal condition. Suppose we call it A^.

If we continue the decomposition up to A^, we have that

R = Aj^+Ag+A^i* . . . +A^; therefore, R is the direct sum of simple left ideals.

Artin proved many more theorems concerning rings satisfying the minimal condition. More of his work will be cited in Chapter V, where we look at "modern" ideal theory. We shall observe then that Artin's paper of 1928 is only the beginning of his contributions to the field of ideal theory. Due to the fact that he was the first to axiomatize the properties of rings satisfying the minimal condition, these rings are known as Artinian rings.

The ideal theory in the period which follows seems to fall naturally into two categories: ideals in noncommutative rings and ideals in commutative rings.

The axiomatizing of the properties of rings satisfying certain properties and the definition of an ideal as an abstract mathematical system, which both emerged during the decade of the 1920's, produced a fusion of the two ideal theories of Kronecker and Dedekind. Emmy

Noether, herself, was a disciple of Dedekind. Van der Waerden, who had been trained in the style of Kronecker, had become the leader in additive ideal theory. However, van der Waerden also became a student of Noether and thus offered no resistance to this fusion. CHAPTER V

IDEALS IN MODERN RING THEORY

The results of Emmy Noether, her students: Grell, Krull and van

der Waerden; Artin and other mathematicians during the decade of the

1920's contributed greatly to the growth of ideal theory and, consequent­

ly to the growth of ring theory in general since its structure theory

rests on Ideals. In view of this fact, ideal theory has become a very

popular subject among ring theorists. The works done on ideals

naturally divide themselves into two categories: those concerned with

ideals associated with commutative rings and those concerned with ideals

in non-commutative rings.

The author, as in the case of the ideal theory developed in the

decade of the 1920's, does not propose to discuss all of the contri­

butions to ideal theory during m o d e m times. The papers and books re­

flecting new results in this field, produced after 1930, are voluminous

but we shall discuss only those papers in ideal theory which are

particularly significant to the development of ideals in modem times.

The author has previously Included a brief biographical sketch

of the particular contributing mathematician in each section. In this

chapter and the following one, we shall discuss only the particular

contribution and its date of publication.

In Chapter III, we discussed an origin of ideals from the polynomial rings of algebraic geometry as opposed to an origin of

233 234 ideals from the study of algebraic number fields which was the basis for Chapter II. There was no such distinction between the results produced during the decade of the 1920's in Chapter IV. This was a result of the fact that the properties generated by the concrete examples, usually in rational fields, were taken over by the abstrac­ tions of the Noether and Artin theories.

All of the work on polynomial ideals, as a distinct discipline, has become a part of commutative ring theory. In fact, algebraic geometry in its most modern form is essentially a study of commutative rings and forms the major portion of the so-called "commutative algebra." Zariski, Cohen, Seidenberg, Krull, Fuchs, Gilmer, Kruse and van der Waerden are among the mathematicians who greatly con­ tributed to this theory.

1. Ideals in Commutative Rings

Some Characteristics of Simple Points of An Algebraic Variety

In 1939 Zariski[221] published a paper which treated systematic­ ally some of the basic questions of the theory of singularities of an algebraic variety. In order to arrive at some of these results,

Zariski made use of some results of Grell[71], Krull[168] and van der

Waerden[205, 207]. We shall see that ideals are the fundamental

"building blocks" of Zariski's study.

Suppose V^ is an algebraic irreducible n-dimensional variety in an affine space Sn(Xj^, Xg, . . , x^) over an algebraically closed ground field K of characteristic zero (see Appendix B); suppose that 235

^1’ ^2’ ' * ’ the coordinates of the general point of and let

be the field of rational functions on of degree of transcendency r over K (see Appendix A). Further suppose we denote by R the ring

^[^1 » ^2’ * ■ * ^n^ whose elements are polynomials in the Thus

the defining ideal of in the polynomial ring

K[x] = K[Xj^, Xg, . . , x^]

is the prime r-diraensional ideal p^ consisting of all polynomials

f(Xj^, Xg, . , J x^) such that

?2’ ' ' "> ^n^ ”

It is true that R is the ring of residue classes and z is the ^r quotient field of R.

It can be shown also that the polynomials in K[x] which vanish at

a point P(a2 ^, ag, . . , a^) of Xg, . . , x^) form a prime

0-dimensional ideal

K ■ *2-^2’ ■ • ■>

The point P is on if and only if p^ c p\ There is a monomorphism between K[x] and K[ç] which sets up a one-to-one correspondence between the prime 0-dimensional ideals p^ in K[x] which contain p^, and

the prime 0-dimensional ideals p^ in

R = K[Ç], where 236

Hence there Is a one-to-one correspondence between the points of in the affine space Sn Xg, . . , x^) and the prime 0-dimensional ideals in R. If PCa^^, a^, . . ^ a^) is a point on then the ideal

Pq = ^2"^2* ^3“^3* • * 1 ^n“^n^ is prime and 0-dimensional and conversely. If P is not on V^, then this ideal (C^^-a^, ^2~^2* ' * ? ^n”% ^ Che unit ideal. p If p^ is any prime 0-dimensional ideal in R, then the ring j- contains a field

K* = K p such that every element of is algebraic over K*. Since K* is ^o isomorphic to K, it is algebraically closed and thus it can be shown that

Hence every element in R satisfies a congruence of the form

U) = c (Pq ) .

Thus we shall say that w has the value c at P. Further

q = a^(p^) and the point PCa^^, ag, . . , a^) on corresponds to p^.

DEFINITION V-1. A point PCa^^, Sg, . . ^ a^) on is said to be a simple point of V^, if there exist r elements rig, . • , in

R such that the ideal

I ” R , D 2* Dg* . . J n^) is divisible by p^ = RCg^-a^^, ^2~^2* ' ' ) ^n”^n^ hut is not divisible by any proper primary ideal belonging to p^. 237

Remarks

1, "Divisibility," as before, means contained in.

2. The condition in the above definition is equivalent to say­ ing that must be an isolated component of the ideal I.

Bearing upon the geometric content of the definition of a simple point, suppose PCa^^, ag, . . , a^) is a simple point of and let

fig» 1I2 » • • > Tlj. he r elements in R such that the ideal p^ =

Çg-ag, . . J Ç^-a^) is an isolated component of the ideal I =

Tig* • • J n^.) ' Without any loss of generality, one can assume that P is at the origin of coordinates where p^ = Çg, . . , . The elements are actually polynomials in the ç's. In each of these polynomials the constant term is zero since each

Hi S O(p^).

Suppose we let

terms of degree - 2. n Zariski[221] shows that the r linear forms Z c..g. are linearly inde- j=l J 2 pendent modulo p . In view of the linear independence of these forms 2 mod p^,. it can be shown that the matrix (c^j) is of rank r. Thus by means of a non-singular linear homogeneous transformation of the coordinates of the general point (see Appendix B), it can be arranged that the elements p^^ have the following form

^2* • * ^ » where i = 1, 2, . . , r and f^^ is a polynomial whose terras are all of degree 1 2. Now if w is any element of p^, then since p^ is an isolated component of I, it follows that there is an element a in R 238

such that a O(p^) and aw = 0(1). Further, since a = c(p ), where 2 c f OeK, then w = 0(1, p^); that is,

w = A^n2+AgP2+ • • • +A^ti^+8, 2 where the A^ER and 3 = O(p^). Now if we replace each by its ex­

pressions given above, that is, = S2 ^+f^(^2 ^, Sg, . . , C^) and ob- 2 serving that 3, as an element of p^, can be expressed as a polynomial

in the ?'s in which all the terms are of degree 1 2, we can now see

that our arbitrary element w in p^ can be put in the form

w = ‘^i^i‘^‘^2^2‘‘’ • * ■ ^2* * * » ^n^ * where g contains only terms of degree ^ 2, in other words, there exist

constants c^^, Cg, . . , c^ such that

“ - • • • Vr(Po)' THEOREM V-1. Let Wg, w^, , . , w^ be elements of p^;

^ 2 = 2 c^jÇ^(p^). A necessary and sufficient condition that p^ bean

isolated component of the ideal (w^, Wg, . . , w^) is that the

determinant be different from zero.

Proof ; = = >

Assume that |c^^| f 0. Given any element w in p^, it is then

possible to find constants d^, dg, . . , d^ such that w = d^w^+dgWg+ 2 2 . . . +d w (p ). Hence p = (w w . . , w , p ) which implies that Tl jT O O .1. IT O p^ is an isolated component of the ideal (w^, w^, . . j w^),

Proof ; < =

Assume that |c^| = 0. There exist constants d^, dg, . . , d^, 2 not all zero, such that d^w^+dgk)g+ . . . d^w^ = O(p^). If, for 2 instance, d^ ^ 0, then it is true that = 0(w^, t*Jg, . . , p^) . 239 2, Thus this congruence, being analogous to = 0(p^, pg, . . , p^^» P^) » leads to a similar conclusion, that is, the ideal Wg, . . , w^) is a multiple of some maximal primary ideal belonging to p^.

Remarks

1. If one takes ?2* • * ï w^, Wg, w^, . . ^ in

^1* ^2’ ’ * ^^n* Chen, for non-special values of the coefficients of these forms, the determinant |c^j| will be different from zero.

2. The elements Çg, . . , have the property that they are algebraically independent and that every element in R is integrally dependent on Çg, . . ,

3. The r elements p^, Pg, . . , p^ have the property that p^ = (Cj^, Çg, . . ) is an isolated component of the ideal (p^,

Pg, . . , p^) and can be chosen in such a manner that every element in

R is integrally dependent on p^^, Pg, . . , p^.

Suppose again we let p^, pg, . . , p^ be r elements in R with the property that p^ is an isolated component of the ideal (p^, Pg,

• • > Hj.). We introduce the ring

K{p} = K{pj^, Pg, . . , p^} of all formal power series in Pj^, Pg, . . , p^ with coefficients in K.

Thus any element in K{p} can be written in the form (see Appendix A)

't'0+*l+ • • • +*m+ • • • where is a form of degree i in p^, pg, . . , p^ , where *=•

. . . +ij;^ and R i s a form of degree H - 1 .

THEOREM V-2, There exists an isomorphic mapping of the ring R onto a subring of K{p} with the following property: if w is any ele­ ment in R and if is the corresponding element in 240

K{p}, then, for all m, “ = = 1'o4l+ - - - +»m Remarks

1. Any element w can be identified with its corresponding power

series ? g. in p . , p „ , . . , p and one writes w = E gj. i=0 ^ i=0 2. Ê gj is said to be the expansion of w in a power series in i=0 ^1’ *^2’ * * * ^r' 3. The Isomorphic mapping of R onto the subring H of K{p^, pg,

. . 5 Hj,} is called a uniformization at P.

4. The elements Pj^, pg, . . , p^ are called uniformizing

parameters.

The above theorem (Theorem V-2), from a geometrical point of view, says that the variety V^ possesses a linear analytical branch at

the point P(0, 0, . . ,0) whose tangent at P has only the point P

in common with the given by the equation = gg = . . . = 0

in the special case where p^ = (see Appendix B).

DEFINITION V-2. P is said to have a uniformization of the whole neighborhood of P when the following conditions are satisfied:

1. There exist in R., r uniformizing parameters p^, pg, . . , p^

for the point P; that is, there exists an isomorphic mapping of R onto

a subring K{pj^, pg, , . , p^} in which every element of p^ is mapped

onto a power series in p^, pg, . . , p^ which has no constant term.

2. The uniformization in 1. has the property that if to an

element w in R there corresponds the power series . . . , then

ti> = (p^^) for all m. 241

THEOREM V-3. If Pg, . , j p^ are uniformizing parameters for a uniformization of the whole neighborhood of the point P(0, 0, . . , 0),

then p^ is an isolated component of the ideal R*(p^, Pg, . . , p^) and hence P is a simple point of V^,

Proof;

By the hypothesis, any element w in R satisfies a congruence of 2 the form w = CQ+c^^p^+ . , , Cq = 0 if u = O(p^) . Hence 2 it follows that p^ = (n^, Pg, • • j P^) and this implies that p^ is an isolated component of the ideal R'(p^, Pg, . . , p^).

The above theorem stated in a stronger form becomes

THEOREM V-4. If there exist r elements p^, Pg, . . , p^ in R such that every element w in R satisfies a congruence of the form 2 Cl) E • • • d’Cj.Pj.Cp^) , then p^ is an isolated component of the

ideal R'(p^, Pg, . . , p^) and P is a simple point of V^.

Remarks

1. The elements p^, Pg, . . , p^ form a modular basis of the p ring —J considered as a K-module. Po 2 2. Since p^, Pg, . . ^ p^ are linearly independent modulo p^,

p then we know that —2. is a K-module of rank r. 2 Po P 3. If is of rank - r, then there exist r elements in R, say Po Pf, Pg, . . , p^ such that every other element w in p^ satisfies a con- 2 gruence of the form w = Cj^p^^+CgPg+ . . . **'c^p^(p^). 242 2 4. = (n^, Hg, • • j Tlj.» Pq ) is an isolated component of the ideal R'(n^, Hg, • • ^ n^),where the Hg, . . , are linearly in- 2 dependent mod p . Po 5. P is a simple point and —j is of rank r.

Pq In view of the above remark, we have the following:

THEOREM V-5. If P is a point on and p^ is the corresponding prime 0-dimensional ideal in R, a necessary and sufficient condition p that P be a simple point is that the K-module be of rank r. Po Now suppose the field I = K(g^, is an algebraic extension of K(G^, ^ • Let m be the relative degree of Z over K(g^, • • Î If w is an element in R, then

N(z-w) = F(z; Gg, . . , = z"*+A^z™"V . . . +A^, where A^, A^, . . ^ A^ are polynomials in gg* • • , Upon re­ ducing the equation F(w; • • » ?j.) “ 0 modulo p^, we find

F(c, 0, 0, . . j 0) = 0, where to = c(p^). Hence c is a root of

F(z, 0, 0, . . , 0). It can be shown that every element in R is integrally dependent on K[S^, Gg, . . , that the ideal R'CG^, gg,

. . , is unmixed and zero-dimensional, and that p^ is one of its components. Let [p^, q^, qg, . . , q^] be the decomposition of the ideal R'(S^, into primary components and let p^, p^,

. . , pg be the associated prime zero-dimensional ideals. Thus we have

THEOREM V-6. Under the assumption that are uniformizing parameters for a uniformisation of the whole neighborhood 243 of the simple point P(0, 0, , , , 0) on and that every element in R is integrally dependent on there exist elements w in R such that (w; Cg, . . , t O(P^), where F(z; ,

ç p is the norm of z-w over K(Ç^, Sg, • • , C^.) • The elements w are characterized by the following condition; if w has the value c at P, that is, if w 5 c(p^) and if p^, p^, . . , p^ are the prime 0- dimensional ideals, other than p^, which divide the ideal R»(g^,

• • } , then (1) f c(p^), where i = 1, 2, . , ^ s.

Remark

The theorem above remains valid if we replace by any other set of uniformizing parameters n^, ng, . . , in R such that every element in R is integrally dependent on n^, Hg, • • ^

Suppose the ideal Z^^, . , , Pj. “ is generated by the different F^ as w varies arbitrarily in R. Further, suppose we denote by Z the h.c.d. of all ideals Z^^^, as n^, n^, . . , vary arbi­ trarily in R only subject to the condition that every element in R be integrally dependent on n^, Hg, . . , n^. We have

THEOREM V-7. If P is a point on and if p^ is the correspond­ ing prime 0-dimensional ideal in R, a necessary and sufficient condi­ tion that P be a simple point is that Z ^ O(p^).

Corollary, The manifold M of singular points of Vj. is algebraic of dimension - r-1 (see Appendix B).

Remark

The manifold M is given by the ideal Z defined above.

THEOREM V-8. Let = . . . ui^Ç„, i = 1, 2, . . , r+1, be r+1 linear forms with indeterminate coefficients u ^ and let 244

F(Ç^, ?2 * • • 5 ^r+1^ = 0 be the irreducible algebraic relation between the Let B' be the ideal whose basis consists of the coefficients f^(g^, ^2 * • • Î 5^) of the various power products of the u^^^ in the polynomial ¥- , where the Ç's have been replaced in the F« by the Çrfl ^r+1 corresponding linear forms in the Ç ’s. The submanifold of V%. defined by this ideal B* is the manifold of singular points of V^.

Suppose we let Pg, • • •> be elements in R such that every element in R is integrally dependent on Pg, • • , and let a^^,

^2* ' ’ * ®r arbitrary constants. The ideal I = R* "^2"^2*

. . j n^-a^) is unmixed and zero dimensional,

I = [qg, qi, . . , = qg^ii - - - ^s' where is a primary ideal belonging to the zero-dimensional prime ideal p^. Let w be an element in R and let G(z; ^2* * * * ~

N(z-w) be the norm of z-w with respect to the field K(n^^, . . , n^).

The following theorem gives a lower bound for the multiplicity of the root Cj^.

THEOREM V-9. If q^ belongs to the exponent p^, then the multi­ plicity of the root c^ is t p^.

In view of this theorem, the question. Do there exist elements w for which Cj is a root of multiplicity exactly pj?, arises. The answer: not necessarily, except when Pj = 1, that is, pj = q^. 3 2 Example : Let pg = UjTi2 be the defining equation of an algebraic

" s ' surface and let ™ . Consider the ring R = q^]. L ' 3 2 Every element of R is integrally dependent on q^, qg, since q^ = q^qg» 245

The ideal I = (n^, Hg) primary, I = q and its prime ideal is

p^ = ri2 » hg, n^) • The exponent of I is 2 because

2 2 Hg “ ~ ^2^2 ' ^3*^4 ~ *^l'^2" On the other hand, the field K(nj_, Hg, Hg, n^) is, in the present case,

of relative degree 3 over K(rij^, Hg) « Since I Itself is primary, it

follows that if w is any element of R and if F(z; n^, Hg) = N(z-m),

then F(z, 0, 0) must have a triple root.

Zarlski[221] also arrived at several results concerning multiple

points of an algebraic variety in his paper. In still another paper,

Zariski extended some of his results on simple points of an algebraic

r-dimensional variety V^.

In 1940, Zariski[220] published a paper which extended some of

the above results. He generalized his results to any defined by a

field Z of algebraic functions over an arbitrary ground field K of

characteristic zero. He did not assume that K is maximally algebraic

in Z. In order to arrive at these results, Zariski made use of some

of the results of van der Waerden[203] and Krull[108] which we dis­

cussed in Chapter IV.

Prime Ideals and Integral Dependence

In 1946, Cohen and Seidenberg[26] published a paper based on some

results obtained by Krull[112]. Krull restricted his research to

commutative rings which were free from zero divisors but Cohen and

Seidenberg simplified his proofs and extended them to commutative rings

which were not necessarily integral domains. In so doing, they made

use of the concept of integral dependence (see Appendix A) which was used in connection with Kronecker's polynomial rings or forms. 246

Since Zariski, and Cohen and Seidenberg drew many of their results from the same paper of Krull[112], the following concept is inevitable.

DEFINITION V-3. Let R and S be commutative rings such that S 23 R and the identity eeR(^S. Let p and q be prime ideals in R and S respectively such that qPlR = p. Then q is said to lie over p.

Remarks ;

1. If over every prime ideal in R there lies a prime ideal in S, we say that the "lying over" theorem holds for the pair of rings R and S.

2. If for every prime ideal t in S lying over p there exists a prime ideal q in S lying over p and containing t, then we say that the

"going-up" theorem holds for R and S.

3. If for every prime ideal q in S lying over p there exists a prime ideal t in S lying over r and contained in q, then the "going- down" theorem will be said to hold.

Cohen and Seidenberg[26] were concerned with the case where S is

integrally dependent on R and prove a series of "lying-over" and "going- up" theorems. As above, we let R and S be commutative rings such that

RcrS and eeR^^S. In addition, suppose that S is integral over R. The

first examination is concerned with whether or not the "lying-over"

theorem holds. The first theorem below is motivated by the following

remark.

Remark

If a maximal ideal in S necessarily lies over a maximal ideal in

R, and if R has a single maximal ideal p, then for the prime ideal p

it is certainly true that there exists a prime ideal in S lying over p;

in fact, it can be shown that every maximal ideal of S will lie over p. 247

THEOREM V-10, Let S be integral over R and let the prime ideal

q in S lie over the prime ideal p in R; that is, q O R = P- Then p is

maximal if and only if q is maximal.

Proof; S R Suppose we consider the residue class rings S* = — and R* = — .

Since q H R = p, then S* OR*. R* and S* are actually integral domains

because p and q are prime ideals and thus S* remains integral over R*.

Our theorem thus becomes:

If R and S are integral domains and S is integral over R, then R

is a field if and only if S is a field.

We observe that in proving this statement of the theorem we must

use the fact that if R is a field, then S is trivially a field. On the

other hand, let S be a field and let a f 0 be an element of R. We must

show that -^R. Since ^ S , it is integral over R and hence we have an

equation of integral dependence:

where the c^eR. Multiplying this equation by a*^ ^ we obtain

^ = -(Cj^+Cgaf . . . +c^a" ^)eR.

Hence R is a field and therefore the theorem is proved. Cohen and

Seidenberg[26] also prove the following additional "lying-over" theorems:

THEOREM V-11. Let S be integral over R. Then for every prime

ideal p in R, there exists a prime ideal q in S lying over p.

THEOREM V-12. Let S be integral over R. Let p be a prime ideal

in R containing the ideal a. If g is an ideal in S such that 3H R = a,

then there exists a prime ideal in S containing 3 and lying over p. 248

THEOREM V-13. Let S be Integral over R and let the prime ideal q in S lie over p in R. Then no ideal in S properly containing q can lie over p in R.

The "going-down" theorems, unlike the "going-up" theorems,require certain assumptions on the zero-divisors of R and S. Even in the case that R and S are Integral domains,further restrictions are required.

The proof of the "going-down" theorem, however, requires two lemmas which Cohen and Seidenberg[26] prove.

LEMMA V-1. Let S be integral over R and let q be an ideal in R.

Then the set of elements in S satisfying an equation of the form

a™+c-a^ . . . +c = 0 1 m is the radical of S*q (see Appendix A).

Remark

Radicals of rings and ideals play a very Important role in de­ termining the structure of noncommutative rings; hence we shall discuss these structures in detail in the next part of this chapter.

LEMMA V-2, Let R be integrally closed in its total quotient ring

R*. If f(x) and g(x) are monic polynomials in R*[x] and h(x) = f(x)g(x) is in R[x], then f(x) and g(x) are in R[x].

We now state and prove the "going-down" theorem:

THEOREM V-14. Let R be an integral domain, integrally closed in its quotient field, S a ring integral over R, with none of its zero divisors in R. Then the "going-down" theorem holds for R and S; that is, if r and p are prime ideals in R with rep, then for every prime ideal q in S lying over p, there exists a prime ideal t contained in q and lying over r. 249

Proof;

Let S* be the total quotient ring of S; now since no element of R is a zero-divisor in S*, then S* contains the quotient field R* of R.

Let D be the multiplicatively closed system in S consisting of elements of the form d6, where der, 6eS, g^q. We consider the set W of ideals in S which contains S«q and do not intersect D. We must show thatW is not empty; that is, we must show that WoS*q.

Claim 1: WoS

Subproof : Suppose d6eS»r. By the first lemma above there exists an equation h(x) = 0 of integral dependence for d6 all of whose coeffi­ cients except the leading coefficient are in r. Let f(x) = 0 be an equation of least degree which is satisfied by d6 over R*. Since the leading coefficient is not a zero-divisor, we may assume that f(x) is monic. Then

h(x) = f(x)g(x), where g(x)eR*[x]. Then by Lemma 2, the coefficients of f(x) and g(x) are in R. Since all of the coefficients of h(x) except the first are in q and h(x) = f(x)g(x), then all of the coefficients of f(x) and g(x), except the first, are in r . Suppose we let

f(x) = x"+Cj^x^ . . . +c^.

Cl n 1 c^ Clearly x’^+Cg— )x + . . . +( ^ ) = 0 is the monic equation of least 1 n degree satisfied by 6 over R*. Similarly as for f(x), we have that

^i ^i ~ d~ since b^d^ = c^^er and d^^^r, we have that b^er. Hence

6^ = -b^6* . -b^Eg'rEq; hence 6eq; this is a contradiction since 6^q. Thus W3S»q. 250

Let t be maximal in W. Thus we have that tflRS S rflRSr, but t^R cannot contain r properly because t does not intersect D. Also t c q since t does not intersect D. We must show that t is prime.

Claim 2 ; t is prime.

Subproof:

Suppose that ydct but yit and 6^t. Since (t, y) and (t, 6) con­ tain t properly, each of them must intersect D; hence their product also intersects D since D is a multiplicatively closed system. This is a contradiction because the product of (t, y) and (t, 6) is in t.

Therefore t is prime. (Note that (t, y) and (t, 6) are ideals generated by t, y and 6.)

Thus by the claims, the "going-down" theorem holds for R and S.

Cohen and Seidenberg[26] show by counterexample that none of the hypotheses of the theorem can be dropped. In each case, the counter­ examples are of a geometrical nature. Working alone, Cohen achieved some more results based on Krull's[106, 112] findings.

The eminent German mathematician Krull, discussed at length in

Chapter IV, was a most prolific researcher in ring theory. He not only contributed to the development of ideal theory during the 1920's but also in this period which followed. In 1938, for example, Krull[106] pub­ lished a paper in which he introduced the notion of a local ring. It was on this concept that Cohen focused his attention in his paper of 1946.

Cohen[25] arrived at some results which characterize the structure and ideal theory of complete local rings. In this paper, he introduced the concept of a generalized local ring and proved several 251 results concerning it. It is very evident in Cohen's study of complete local rings that ideals are the principal structures in the analysis and characterization of these rings.

Primal Ideals

In Emmy Noether's[1631 study of decomposition theorems, the emphasis was on prime ideals and their associated components. Fuchs

[54] was Interested in a certain type of decomposition whose components are associated with the maximal prime ideals which he called primal ideals. He published a paper about them in 1950. Fuchs[54] defined his primal ideals by utilizing the concept of "relative primeness."

DEFINITION V-4. An ideal a is said to be a primal ideal if the elements which are not prime to it form an ideal p called the adjoint ideal of a. (An element a is prime to g if abeg implies beg for an ideal g.)

Remarks

1. a is a primal ideal and p is adjoint to a if and only if aben and b^a imply asp and conversely if aep, there exists always an element b not in « such that abea.

2. The elements non-prime to a represent, in the residue class ring — , the zero factors.

The following theorem shows the relationship that exists between irreducible ideals and primal Ideals.

THEOREM V-15. Every irreducible ideal is primal.

Proof;

Suppose that i is an irreducible ideal and that a^, a^ are elements which are not prime to i. Then i:(a^) and i:(a2 ) are proper 252 divisors of i, hence i: (a^) Hi; (a^) = l((a^)+(a2 )) and i((aj^)+(a2 >) f i.

Thus the element a^^-a^ cannot be prime to i. Hence the elements not prime to i"form an ideal. Therefore i is primal.

THEOREM V-16. The reduced intersection of a finite number of primal ideals

a = . . .

By hypothesis the intersection a = is re­ duced. By the developments of E. Noether[163] we can conclude that an element a is not prime to a if and only if it is not prime to at least one of the a^; that is, if and only if a belongs to at least one of the p^. This implies that a is primal if and only if the union of the elements of all p^ is a prime ideal. Hence sufficiency is clear.

Proof ! <==

We must show that the union of the elements of a finite set of primes is not a prime ideal again unless one divides the others.

Suppose we let Pj» P2 » • • ; P^ denote the maximal prime ideals of the p^y Pg, . . j p^; that is, those ideals which are divided by no dif­ ferent one in the set of Pj^, Pg» • • % P^; further, we can assume that k > 1. Now since none of the p^ divide one another we can therefore find elements r^ (& = 1, 2, . . , k) contained in p*rip*n . . .

O p J ^ l H . . . n p ^ but not in pJ. If a were primal, the fact that the

are not prime to a would imply that r = . . . +r^ is not 253

prime to a. Consequently r would belong to one of the r^ and so r would belong to one of the p^. This is a contradiction since each r^

except r^^ belongs to p*. Hence necessity is clear.

The theorems which follow summarize the decomposition of primal

Ideals. They are proved by Fuchs[54].

THEOREM V-17. Every ideal is representable as the intersection

of its primal divisors.

THEOREM V-18. In a ring in which each ideal may be represented

as the intersection of a finite set of irreducible ideals, each ideal

possesses a normal decomposition into a finite number of primal ideals.

THEOREM V-19. In two finite normal primal decompositions of an

ideal the numbers of the components as well as their adjoint prime

Ideals are necessarily the same.

We close our discussion of primal ideals by giving an example

which will establish the fact that primal decompositions do not coin­

cide with the other decompositions which Noether[163] utilized. They

are the (1) irreducible, (2) primary, (3) quasi-primary, (4) relatively-

prime-indecomposable». and the (5) direct-indecomposable ideals. Suppose

we consider the polynomial ring of x, y, and z; the decompositions of

(2) and (3) of the ideal , 2 2 2. Ü) = (x y, xy z )

are

(2) w = (x) n (y) n (x^, y^)n(x^, z^)

with (x), (y), (x, y) and (x, z) as associated prime ideals;

(3) 0 1 = (x^, xy^z^)iA (y) 254 with the radicals (x), (y). A normal primal decomposition of w is 2„ 2,^ , 2 2, (11= (x y, xy ) n (x , z )f the adjoint prime ideals being (x, y) and (x, z). In this case

(11 = (x^, xz^) n (x^y, y^) ^ is normal primal with radicals (x), (y) and with the adjoint prime ideals (x, z), (x, y) . oi is a relatively-prirae-indecoraposable ideal.

Some Results Based on A Classical Theorem of Noether in Ideal Theory

Emmy Noether[162] stated and proved the following theorem which became a classic in ring and ideal theory:

THEOREM V-20. An integral domain D with unit is a Dedekind do­ main if and only if D is Noetherian, of degree less than two, and integrally closed.

Noether, in her original form of this theorem, did not require that D contain a unit element. However, by making certain restrictions on the prime ideal factorization of each ideal, she showed that D must contain a unit element.

Gilmer[66] considered an integral domain J with this property:

Every ideal of J may be expressed as a product of prime ideals. Gilmer observed that an integral domain J with the above property need not contain a unit element. Further, he observed that the factorization of an ideal as a product of prime ideals is unique and hence J is Noether- ian, of dimension less than two, and is integrally closed. However, it is true that an integral domain without unit having these three properties need not satisfy the property above. In Gilmer's investiga­ tion, he used the following notation: J is an integral domain such that 255 every Ideal of J is expressable as a product of prime Ideals, k is the quotient field of J, J* will denote the subring of k generated by J and the unit element e of k, D is an Integral domain not necessarily con­ taining a unit, and D* is a fractionary ideal of D, where a fractional / ideal is a D-submodule of the quotient field of the integral domain D whose elements admit a common denominator d. DEFINITION V-5, A fractionary ideal F of D is said to be invertible if F has an inverse when considered as an element of S, the collection of all nonzero fractionary ideals of D.

Remarks

1. A nonzero principal fractionary ideal is invertible and

(d)"^ = (j).

2. A product of fractionary ideals is invertible if and only if each of the factors is invertible.

Zariski-Samuel[222] proved the above remarks as well as the two lemmas which follow:

LEMMA V-3. If A is an invertible fractional ideal of the integral -1 domain D, then A = D*;A. Also A has a finite module basis over D.

LEMMA V-4. Suppose A is a proper ideal of the domain D such that

A may be expressed as a product of invertible prime ideals of D. This representation is unique if DSD*, or is unique within factors of D if

D = D*.

In the theorems which follow, J will denote an integral domain without a unit such that every ideal of J is expressable as a product of prime ideals. 256

THEOREM V-21, Every nonzero proper prime ideal of J is invertible and maximal.

Proof :

Suppose first that there exists a nonzero proper invertible prime ideal P of J such that P is not maximal. We choose a such that

PcP+(a)cJ. 2 We express P+(a) and P+(a ) as products of prime ideals;

P+(a) = . . . P^, P+(a^) = . . . Qg, where each P^ and each is a proper ideal of J. In J = ^ we have;

(â) = j^Pj^ . . . P^, (â)^ = . . . Qg. By Lemma 4, s = 2r and by the proper indexing P^ = " Q2i'

J does not contain a unit element then statement 2 implies also that t = 2k so that P+(a^) = [P+(a)]^. If J contains a unit, then (a) = J^Pj^

. . , P^, so that r is positive and (a) = P^ . . . P^. In the same fashion (a)^ = . Q^. Therefore [P+(a)]^ = P^ . . . P^ = P+(a^). 2 2 For either case, therefore, P+(a ) = [P+(a)] . This Implies that

Pc[P+(a)]^cP^+(a). 2 Thus any element xeP can be written in the form x = y+za with yeP and zeJ. Then we have that zaeP which implies that zeP since a|éP; that is, 2 2 P c P +Pg. But trivially P ^ P fP^. These results imply that

P = P^+Pg = P(P+(a)).

Since we assumed that P is invertible, the P ^exists and we can multiply -1 by P to obtain that

J = P+(a). 257

Since a is an arbitrary element of theicomplement of P in J, then P is maximal. We must next show that every proper prime ideal P in J is invertible. We let a nonzero element beP and set

(b) = P^Pg . . . Pg. where the P^ are prime ideals. Since P contains P^P^ . . . P^, it con­ tains some P^. But by Zariski-Samuel[222], every P^ is invertible since P^P^ . . . P^ is invertible. Thus every P^ is maximal by the first part of the proof. Since P contains one of them, say P^, we have

P = and hence P is invertible.

THEOREM V-22. J is a Noetherian domain.

Proof:

Suppose J contains a proper nonzero prime ideal P ; then P = (p^^, pg, . . f Pg) is maximal and finitely generated by Theorem 21 and

Lemma 3. Therefore, if deJ and d^P, then J = (p^, . . , p^, d). If Ic (0) Is the only proper prime ideal of J, then given deJ, d f 0, (d) = J for some integer k t 1. Then J is invertible and hence finitely generated. This implies that every prime ideal is finitely generated.

But J has the property that every ideal of J is expressible as a product of prime ideals. Hence every ideal of J is finitely generated; thus J is Noetherian.

Gilmer[66] also proved these two theorems which we state without proof;

THEOREM V-23. Every nonzero ideal of J is a power of J and J is a principal ideal domain.

THEOREM V-24. J* is a discrete valuation ring (see Appendix A) of rank one. Conversely, if D is a discrete valuation ring of rank one 258 with minimal ideal M and if D = M*, then M is a domain without unit such that every ideal of M is expressable as a product of prime ideals.

In view of the results that Gilmer proved concerning the theorem of Noether, we are reminded of her influence on modern ideal theory.

Some Fundamentals of Rings in Which All Subrings Are Ideals

Earlier in our discussion of Ideals, we mentioned the fact that an ideal plays the role in ring theory that a normal subgroup plays in group theory. A group G in which every subgroup is normal is called a

Hamiltonian group. In 1968 Kruse made a study of rings in which every subring is an ideal.

Kruse[114], in analogy with a Hamiltonian group, defined a

Hamiltonian ring:

DEFINITION V-6. An associative ring in which every subring is a

(two-sided) ideal is called a Hamiltonian ring or simply an.H-ring.

In order to investigate the structure of such rings, certain pre­ liminaries in terms of definitions and basic results must be established.

DEFINITION V-7. The characteristic of a ring R is the least natural number N for which Ni^ = 0 for all i|t£R provided such an N exists.

Remark

If such an N does not exist, the characteristic is zero.

DEFINITION V-8. The exponent of a nilpotent element <}) is the least integer r for which (|i^ = 0.

DEFINITION V-9. An H-ring whose additive group is a p-group is called a p-ring where p is a prime; such rings are sometimes called

H-p-rings. 259

KruseI114] utilizes the following known results which we list as lemmas without proof:

LEMMA V-5. The Jacobson radical of an H-ring is nil.

LEMMA V-6. A semi-simple H-ring is isomorphic to a ring direct sum of the form

NQ e Ee F , PeB where B is a set of primes, Q is the ring of rational integers, NO the subring generated by an Integer N, F^ is the prime field of order p, and each prime p divides N.

LEMMA V-7. Let m be a nilpotent element of characteristic 0 in an 2 3 H-ring. Then w has a nonzero square-free characteristic and w = 0 .

LEMMA V-8, Every subring and every homomorphic image of an H-ring is an H-ring.

LEMMA V-9. A ring is an H-ring if and only if every subring generated by a single element is an ideal.

LEMMA V-10. A ring R with torsion additive group is isomorphic to a restricted ring direct sum of p-rings R^ for different primes p.

LEMMA V-11. R is an H-ring if and: only if each R^ is an H-ring.

DEFINITION V-10. A subring S of a ring R almost annihilates R if, for all (j>eS,

1. ^ = 0, 2 2. M(J) = 0 for some square-free integer M which depends on

3. (|»R ^ and R S {(|)^}. ({(|)^} is the additive subgroup 2 generated by (j) .)

Remarks

1. If S = R, the ring R is called almost-null. 260 2 2. When the subring S is a nil H-ring, then M(|) = 0 for some 3 square-free integer M which depends on (|> implies that (j) =0.

3. All almost-null rings are nilpotent H-rlngs.

4. Suppose the subring'S almost annihilates the ring R. Choose (|ieS. 2 Then annihilates R if and only if <|i = 0 .

Some further basic results again listed as Lemmas without proofs

are; 3 2 LEMMA V-12. If 4> is an element of a nil H-p-ring, then <(> 2 2 where {4> } is the additive subgroup generated by .

LEMMA V-13. Let R be a nil H-p-ring and let <|)eR, ÇeR, Then

there exist integers U, U*, V, V such that

4>C = = U'ç+V'ç^.

LEMMA V-14. Let 4' and w be elements in a nil H-ring. Suppose

2 2 that M = char 4" f 0 and M divides char tt). Then char is square-free.

LEMMA V-15. A nil p-ring of characteristic 0 is an H-ring if and

only if it is almost-null.

LEMMA V-16. A nil ring which contains an element of characteris­

tic 0 is an H-ring if and only if it is almost-null.

PROPOSITION 1, For a ring R and a prime p define

Rp - {4> e RI p*^ - 0} .

A necessary and sufficient condition that R be almost-null is that

R = ^pRp and that each subring R^ satisfies one of the following con­

ditions. (N is the annihilator of R .) P P 1. R = N is null. P P 2. Rp = {eNp, 4» eN^ and char ^ = p. 261 2 2 3. Rp = {4>, Ç}+Np, where p4», pg, <|) eNp, char 4* = p, and where 2 2 2 there are integers A, F, and F' such that g = A<(> , (fg = F(|) , g(j) = 2 F'4t , and for which the congruence

X^+X(F+F')+A = O(mod p) has no integer solution X.

Proof ; = >

To establish necessity, suppose that R is almost-null and define

R = {4>eR|p(|)^ = 0}. 2 By definition of a subring almost annihilating a ring, M(() = 0 for some square-free integer which depends on 4>. This implies that

R = ZR , where the restricted sum is taken over all primes p. If the P P subring Rp satisfies neither (1) nor (2) of the conclusion, then there exist elements 4»ERp, gcR^, with 4>, ? linearly independent mod ^pRp,

Np> ; this implies 0, and (C+X4>)^ ^ 0 for any X; otherwise, ge{(f)}+Np. It is necessary to prove that

Now again by definition of a subring almost annihilating a ring,

4>R S {4 *^} and R4> S (4)^}; hence if 4"S f 0 or C4> f 0, then {4>^î = otherwise the fact that 4>(4'+?)e{ implies that {4>^J = Now 2 (Ç+X4>) ^ 0 is equivalent to saying

X^+X(F+F')+A = O(mod p) has no integer solution X. Hence if Rp = {4», 5)+Rp, then (3) is true.

Suppose, on the other hand, that there exists some 4'ERp, 4'e { 4', Ç}+Rp.

Hence {4>^J = (g^l = (4'^K But by Remark 4 to the definition of a sub­ ring annihilating a ring, we have that 262

(X(|)+YÇ+Z4()^ = 0

Implies that

X = Y 5 Z(mod p). 2 (The existence of a nonzero solution of (X<(i+Yg+Zi|j) = 0 follows .from the fact that every quadratic form in three variables over a field of p elements represents zero.) This contradiction establishes necessity.

Proof : < =

Suppose that (1), (2), and (3) of the conclusion of the proposition hold. Then sufficiency follows by straightforward verification.

Hence our proposition holds.

Lemma 6 gives the structure of semi-simple H-rings. This state­ ment along with some others will be used to obtain the structure of general H-rings. In order to do this three cases must be considered:

1. H-rings which contain no elements of characteristic zero.

2. H-rings whose semi-simple parts contain an element of characteristic zero.

3. H-rings which contain a nilpotent element of characteristic zero.

Lemma 10 reduces case 1 to the study of H-p-rings. The following proposition reduces case 1 to the study of nil H-p-rings and is proved by Krull[114].

PROPOSITION 2. A ring is an H-p-ring if and only if it is isomorphic to a ring R which satisfies one of the following conditions:

1. R is a nil H-p-ring.

2. R = F 0 N, where F is the field of p elements and N is a nil

H-p-ring. 263

0 n 3. R = ----- is the ring of rational integers modulo p , n > 1. (pnj R In case 2, we let be the radical of R. By Lemma 6,^ is isomorphic to

NQ 0 Q (M) for some positive integer N and square-free positive integer M which 2 divides N. Thus there is an element veR such that v -N = ilieR and V ^ char V = 0. Thus

PROPOSITION 3. vi|) = i|(V = = 0. C = char 'i' 0 is square- free and divides N.

Proof : 2 Suppose that C = 0. Let K = char 4* . By Lemma 7, K f 0. Then

Kv(2Ky) i (2Kv). This implies that C # 0. Suppose there is a prime p 2 C C such that p divides C. Then ^(—)v f ((—)V), Thus, C is square free.

Now choose a prime p which does not divide N. The ring — ^ ^ must (Pu* ) 2 satisfy Proposition 2 and this implies that v i |j = ij(V = O(mod (pv* 4' )).

This must hold for infinitely many primes p, and thus \>ip = titv = O(mod

(4^^)). Hence

v(CV+4;)e(Cv+4') (mod (4»^)) ; 3 this implies that C divides N. Finally Lemma 12 implies that 4" = 0 , which, with 4"^ = (v^-Nv)4' = v(u4'), implies 4*^ = 0. Thus also v4> = 4^^

= 0. Hence the proposition is proved.

PROPOSITION 4. Ng. contains no element of characteristic 0.

Proof ; 2 Suppose that meN and char m = 0. By Lemma 7, K = char m f 0.

Let g = 2C\)+KW. Then (Ç) . 264

Remarks

1. In addition to the subring of characteristic 0, the semi- R Q simple ring ^m a y contain a torsion subring isomorphic to when M is a square-free integer which divides N,

2. If M f 1, then there is an idempotent esR such that

M s ev = ve = O(mod N). e PROPOSITION 5. Me = evi = ve = 0, e annihilates N_. u Proof :

Let Q = char e- By the preceding proposition Q f 0. Suppose for some prime p that p^ divides Q. Let * = ONv+(^)e. Then eijiE -((j)).

Thus Q is square-free,,w h i c h implies that (e) contains no nilpotent elements, so (e)AN = this implies that e annihilates N •

In order to complete the determination of the structure of R, we must take a closer look at the structure of N.

PROPOSITION 6. Np = 0, unless p divides N,

Proof:

For Np ^ 0 choose 0 4 4>^Np, with 4'^ = p4* = 0. Thenv(p.v+4>)e

(pvf4) ) implies p divides N.

PROPOSITION 7. N is almost-null. P Proof ;

Let 4»ENp. Let char 4> = p^. Then 4'(p4>+p®\')£ (p4>+p®v) which implies that p4>^ = 0. Let C^Np and let p^ = max (char 4»* char S) .

4'(g+p*'v)e (g+p*"v) 2 2 implies 4>ge{g }. 4)Çe {4> } is dual (see Appendix A). Thus Np is almost- null. 265

By writing 4> = where ippSNp, we proceed to Investigate the relation between v and each N^. First we observe that p^/^ = 0 and ijjp = 0 unless p divides C. Further we have

PROPOSITION 8. The element v may be chosen in such a way that

and Np satisfy one of the following conditions:

1. tp = 0, N annihilates v, and char N divides N. P P P 2. ijjp 4 0, Np annihilates Nq, and char Np divides N.

3. Ü) f 0, N = {(b }+M , where 4< eM , p4> eM , M annihilates R, P P P P P P P P P char M divides N, and {i|( } contains 4* o* Nt|> , v4> » and 4>_v* The P P P P P 2 equation (v+X4> ) = Nu+ Z 4> ■’ qfp 2 has no solution X. Finally, if 4>p “ 0» then p = 2 and

N4» = v4i =

«O' N- = N^ 0 E 0 F (restricted) , " " peB P where N is almost-null, B is a set of primes, and F^ is the fieldof p elements.

Finally, case 3. the H-rings which contain a nilpotent element of characteristic 0, is characterized by the proposition below:

PROPOSITION 10. A ring which contains a nilpotent element of characteristic 0 is an H-ring if and only if it is isomorphic to a ring

N_ = N^ 0 S 0 F (restricted) " ^ peB P 266 where N is almost-null, B is a set of primes, and is the field of p elements.

Thus the class of almost-null rings is of fundamental importance in the determination of Hamiltonian rings,

2, Ideals in Noncommutative Rings

The main focus in noncommutative ring theory is structure theory.

The primary objective of any aspect of structure theory is to describe some general object in terms of some simpler ones. As one would expect, the simpler objects in ring theory are ideals. A very convenient method of studying structure theory in noncommutative rings is via special ideals called radicals.

There are several notions of a radical in ring theory. We shall discuss some of these notions of a radical in the ensuing sections of this chapter. We shall see that semi-simple (see Appendix A) rings are characterized by the fact that their radicals are the zero ideal

(0). Another important property of radicals is that if R is a ring such that the radical of R, rad R, is not the zero ideal, then the radical of R(mod rad R) is the zero ideal, that is, the radical of the quotient ring ^ is the zero ideal. Thus we have a reduction of any ring to a semi-simple ring. Hopkins worked primarily with the so- called "classical" radical (see Appendix A).

Some Aspects of Rings with Minimal Condition on Left Ideals

In 1939 Hopkins[90] published a paper in which he examined the structure of a ring R with radical, rad R 4 (0) and such that R has the 267 minimal condition on left ideals (MLI) (see Appendix A). The

(classical) radical of a noncommutative ring has been defined in various ways. Among them are (throughout this section we shall assume that R has the minimal condition on left ideals):

DEFINITION V-11. The radical of a ring R is defined to be the maximal two-sided nil-ideal (see Appendix A).

DEFINITION V-12. Suppose W is the sum of all nilpotent ideals in a ring R. If W itself is a nilpotent ideal, then W is called the classical or nilpotent radical of R written rad R.

One of the most important results attained by Hopkins was to show that the radical of R is nilpotent. (Hopkins utilized Definition

V-11.) However, we require some preliminary theorems, which we shall state as lemmas, before we discuss this important result and highlight of this section. (The proofs of these lemmas are given in Hopkin's work which was cited above.)

LEMMA V-17. If is any minimal nonzero left ideal of a ring R and n is any left nil-ideal of R, then

nAj^ = (0).

LEMMA V-18. If R is a MLI ring with rad R 7^ (0), then the right annihilator Nj^ of rad R in rad R is a nonzero two-sided ideal of R.

LEMMA V-19. If s^, s^, . . , s^ are any n subrings of a ring R, and if the symbol s:t is defined to be the set of all elements x in R for which tx is contained in s, then (. . . (Sj^:s2 ):s2 . . . ):s^ =

* 268

THEOREM V-25. The radical of an MLI ring is nilpotent.

Proof:

Let R be an MLI ring with rad R f (0), From Lemma 18, we know that the right annihilator of rad R in rad R is a nonzero, two-sided ideal. If

= rad R, then

(rad R)^ = (rad R)N = (0) and hence we're done. Thus we assume that N^crad R and construct the quotient rings

, and rad R (rad R)' =

Kothe[104] earlier proved that every quotient ring of an MLI ring is a

MLI ring. Hence R' is an MLI ring with radical (rad R)'o(O). By

Lemma 18, we know that (rad R) contains a right annihilator (0)' which is a two-sided ideal of R'. Let denote the homomorphic image of Ng in R under the homomorphism R-»-R'. Since Ng ^ (0) ^, we have

Suppose that and have already been defined for i< j. R R Then we define N' to be the right annihilator of »--- in-»--- and N. J *j-l ^j-1 J to be the homomorphic image of in R. Now consider; the ascending chain of ideals (annihilators) (see Definition 14 below) (1) N^ S N2 £

. . . ; then by definition

NÎ = (0):(P^) ^ "i-1 269 and

= N^_2:(rad R)'.

By Lemma 19, we observe that N^_^:(rad R) = (. . . ((0):radR) . . . ) rad R = (0):(rad R)^. Hence = (0);(rad R)^. Now if we consider the 2 descending chain of ideals (2) (rad R) 2 (rad R) 2 . . . . Each term in (2) is a left ideal of R and since R is a ML I, we must have the equality sign in (2) after a finite number of terms. Let k be the k k+1 smallest exponent for which (rad R) = (rad R) = . . . , Now

N^crad R implies that N^^^D(O) and N^^^^k ~ (0):(rad)^ and

1r+1 If If+l = (0) : (rad R) and since (rad) = (rad) we have that N^ = N^_^^ which implies that N^ is equal to rad R. From N^ = rad R, we have that

(rad R)k+1 = (rad R)^N^ = (0); that is, rad R is nilpotent and of exponent p = k+1. Thus our theorem is proved.

There are two important corollaries which are direct consequences of the above theorem of Hopkins; however, it is necessary that we state the definition of a semi-primary ring.

DEFINITION V-13. A ring R is called semi-primary if the quotient- g ring with respect to rad R is semi-simple; that is ^ is semi-simple.

Corollary 1. An MLI ring is semi-primary.

Corollary 2. In an MLI ring, any subring containing only nil- potent elements is itself nilpotent.

It is not difficult to establish the next theorem if rad R is nilpotent and of exponent p = k+1. Though the last theorem is the most important one of this section, we Include another related theorem which is also important. A definition is required first. 270

DEFINITION 7-14. The left annihilator of rad R in rad R is the set of all elements x in rad R for which

x(rad R) = (0).

Remark

The analogous results may be obtained for right annihilator, minimal right ideal, etc.

THEOREM V-26. Let R be an MLI ring with rad R 5^ (0). If is defined to be the left annihilator of (rad R)^ in rad R, then c for i < p-2; Mp_^ = rad R, where p is the exponent of rad R; and each

M^ is a two-sided ideal of R.

Proof ;

Since M^ contains (rad R)^ which is not the zero ideal, then

can not be zero for ill. But M^_^ does not contain (rad R)^"^; if it did, we should have

(rad R)P"l(rad R)^“^ s M^_^(rad R)^~^ = 0 , whereas

(rad R)P"^'(rad R)^“^ = (rad R)^“^ f 0 .

Hence M^_^ is properly contained in R. Hence M^^^ “ rad R. Thus our theorem is proved.

Remarks

1. By its definition Mj^ is a left ideal for each i.

2. From the relations

(M^R)(rad R)^ = M^(R(rad R)^) 5 M^(rad R)^ = (0), it is equally true that M^ is a right ideal of R,

3. ^i+1 is the left annihilator of in î that is. Mi M^ ^i

Mi^i(rad R) • = M^. 271

Hopkin's[90] outstanding results about the classical radical and

MLI rings were followed by Baer's "radical ideals" in the development of modern ideal theory. Before we look at Baer's contribution, how­ ever, we should like to indicate some of the general properties of radicals.

In view of the fact that there are several generalizations of a radical and the fact that we shall encounter several of them in our development of modern ideal theory, we pause here to inspect a list of the desirable properties of a radical as summarized by Gray[68],

Radical Properties:

1. The radical exists in every ring.

2. If N is the radical of R, then the radical of — is the zero N ideal of ^ . N

3. The radical of an ideal is the Intersection of the ideal and the radical of the whole ring.

4. The radical also contains certain one-sided ideals.

5. The radical of a matrix ring over a ring R is the matrix ring over the radical of R.

We shall refer to this list of properties in our discussion of other notions of a radical.

Some Fundamentals of Radical Ideals

Baer, like Hopkins, made use of the work of Kothe[104] and others in research on rings and ideals. In 1943, Baer[10] published a paper in which he defined a radical ideal and proved several results based on this algebraic structure. The result of most importance in his work was the lemma below. 272

LEMMA V-20. The sum N = N(R) of all the nilpotent right ideals in R is a two-sided nil-ideal in R.

DEFINITION V-15. An ideal B in a ring R is called a radical ideal if

1. B is a two-sided ideal.

2. B is a nil-ideal.

3. The quotient ring ~ does not contain nilpotent right-ideals different from (0).

Remarks

1. There exist ideals in R which satisfy only 1. and 2. of the above definitions.

2. It is possible to form the sum U = U(R) of all ideals in R which satisfy only 1. and 2. of the given definition of a radical ideal,

3. The ideal U is a two-sided ideal in R.

4. There are also ideals in R which satisfy 1, and 3. of the definition (B = R is an example).

5. It is possible to form the g.c.d. L = L(R) of all the ideals

B in R which satisfy 1. and 3. (The g.c.d. is the greatest common divisor.)

6. The ideal L is a two-sided ideal in R.

DEFINITION V-16. The two-sided ideal U which is the sum of all ideals B in R satisfying 1. and 2. of the definition of a radical ideal is called the upper radical of R.

DEFINITION V-17. The two-sided ideal L which is the g.c.d. of all ideals in R satisfying 1, and 3. of the definition of radical ideal is called the lower radical of R. 273

The following discussion is not restricted to rings satisfying

the minimal or maximal conditions;

THEOREM V-27. The upper and lower radicals of the ring R are

radical ideals in R.

Proof :

K6the[104] proved that the sum of all the two-sided nil,ideals in

R is a nil-ideal. Suppose we denote by W the uniquely determined two-

sided ideal in R which satisfies

yR K where N(^) is the sum of all nilpotent right ideals of — . By Lemma

20, there exists for every element x in W a positive integer i such

that x^ is an element in the nil ideal U, and thus every element in W

is a nil element. The definition of the upper radical implies that

W = U, N(^) = (0). Thus the upper radical of R is a radical ideal in

R. Now it is true that L « U by definition of L. Since the upper

radical U is a radical ideal and since the lower radical L is cer­

tainly a part.of every radical ideal in R, then L is a nil-ideal since

U is a nil ideal. Every nilpotent right ideal in — is of the form ^ L L for a suitable right-ideal S between L and R, Mow since is nilpotent,

then there exists a positive integer i such that S^ 5 L. Suppose T is a two-sided ideal in R satisfying 3. of the definition of a radical

ideal. Thus

(T+S)^ 5 T+S^ - T+L = T

Implies that L - T and i = o. Then by 3. = (Q) or T+L = T V fp / T or S S T. Hence S is a part of every ideal T which satisfies 1. and 3.

Hence S 9 T; thus L is a radical ideal in R. 274

Remarks

1. If T is a two-sided ideal between L and U, then the upper radical of — is — . T T

2. If T is a radical ideal; then the lower radical of f = (0).

3. Every subideal of the upper radical is a nil-ideal.

4. The upper and lower radicals of a ring R are equal if E is a nilpotent ideal in ^ .

5. If the upper and lower radicals are equal, then this ideal may be termed the radical K = K(R) of the ring R, and we say that the radical of R exists.

6. Baer[10] demonstrates that other radical ideals exist by constructing a ring with the following properties:

a. Every element in the ring is a nil element so that the

ring is its own upper radical.

b. The ring does not contain nilpotent right ideals dif­

ferent from the zero ideal so that its lower radical is the zero

ideal.

c. The ring contains a two-sided ideal which is not a

radical ideal.

Baer[10] proved two main results which describe the upper radical of a ring.

THEOREM V-28, If T is a radical ideal in the ring R and if every right ideal different from the zero ideal (0) in the quotient ring Y contains a minimal right ideal, then T is the upper radical of R. 275

Proof:

Let J be a nil ideal in R, If J in not part of T, then there V exists an ideal V between T and T+J such that ^ is a minimal right ideal in — . Now since — is a part of (T+J) and since J is a nil ideal, T T T — is a nil ideal. But since a minimal right ideal is either nil- T potent or idempotent, and since T is a radical ideal in R, then — is T idempotent. But by van der Waerdèn[205], idempotent minimal right ideals contain idempotent elements which are not 0. Thus the nil ideal X contains an idempotent element which is not 0; this is a con­ tradiction. Hence every nil ideal in R is contained in T, But since

T is a radical ideal, it is a two-sided nil ideal and is therefore part of the upper radical. Since the upper radical is the sum of all the two-sided nil ideals, then V = T contains every nil ideal.

DEFINITION V-18. The element e in R is a left-identity element if ex = X for every element x in the ring R.

l e m m a . V-21. If the ring R contains a left identity element e, then each nil ideal is a part of every maximal right ideal.

THEOREM V-29. If the rigg R contains a left-identity element and if every right ideal different from (0) in the quotient ring & contains a minimal right ideal, then the upper radical U of R is the g.c.d. of all the maximal right ideals in R.

Proof:

Let J be the intersection of all the maximal right ideals in ^ .

If J were different from (0), then J would contain a minimal right ideal J'. By Oray[68], there exists an idempotent e f Ü in J*. But — 276 is the direct sum of the right idealsJ’ and Z, where J' consists of all the elements v = ev and where Z consists of the elements z satisfying ez = 0, Hence the g.c.d. of Z andJ' is (0). Since J' is a minimal right ideal in — , Z is a maximal right ideal in — . Thus J' S J S Z; U U this is a contradiction showing that J = 0. Hence Ü is the intersec­ tion of all the maximal right ideals in R which contain U; and by

Lemma 2, U is a part of every maximal right ideal in R, because U is a nil ideal by Theorem V-27.

LEMMA V-22. If N is a sum of (a finite or infinite number of) minimal right ideals in the ring R,then every right ideal contained in

N is a direct summand of N and is itself a direct sum of minimal right ideals in R.

Due to Baer, we have the following definition of a two-sided ideal associated with his radical ideal in a ring.

DEFINITION V-19. The sum M = M(R) of all the minimal right ideals in the ring R is called the anti-radical of R.

Baer was successful in obtaining several results for .the anti­ radical of a ring, anti-radical series, and powers of radical ideals.

Many of his arguments involve transfinite induction. Unfortunately, we must omit these from our discussion due to their long lengths and lesser Importance in our study. However, we do invite the interested reader to puruse them in the original paper, which was cited earlier in our discussion. However, we shall mention the fact that when the upper and lower radical are identical; that is

U(R) = L(R) then the Baer radical. Ill

B(R) = U(R) = L(R) exists. Further, if the ring R is either left Artinlan or right

Artinian, the Baer radical exists and coincides with the classical radical.

There are several generalizations of the classical radical as we mentioned earlier. We shall see that these generalizations were invented to do for general rings what the nilpotent radical does for rings with the minimal condition on ideals. Levitzki invented such a radical.

On the Radical of a General Ring

Levitzki[139] in 1943 published a paper in which he introduced another notion of a radical. He made use of ideals which he called semi-nilpotent.

DEFINITION V-20. A right ideal is called semi-nilpotent if each ring generated by a finite set of elements belonging to the ideal is nilpotent. (A right ideal which is not semi-nilpotent is called semi- regular.)

Levitzki proved the results which follow:

THEOREM V-30. The sum R = R^^+Rg of two semi-nilpotent right ideals R^ and R^ is a semi-nilpotent right ideal.

Proof :

Suppose R is a semi-regular ideal; that is, R contains a finite set

=01' =02' ' * ) ’^On that the ring S* = is potent (not nilpotent). We write r^^^ = ^ol^yl* *^ere r^^ER^, ^yl^^2* then also S** = {^oi* • • } =0 %, j potent, since

S* 5 S**. Define the set r^, tg, . . , r^ as follows: (1) each r^ is 278

either an r^^ or an r^^, (2) the ring {r^, rg, • • » r^} is potent,

(3) each ring generated by a proper subset of the is nilpotent.

One obtains such a set by suppressing one way or another the greatest

possible number of elements of the set r^^* , r^^, . . , r^^

so that the remaining set still generates a potent ring. The set of

the r^^ necessarily contains certain r^^ as well as certain r^j^ because

the r^^ as well as the r^^ taken separately generate nilpotent rings;

hence ra > 2. Now consider the rings T = {r^, r^, . . , r^} and U =

{rg, r^, . . , r^}. By the definition of the r^, it is true that T is

potent while U is nilpotent. Denote by ethe Index of the nilpotent

element r^ and by a the index of the nilpotent ring U; denote by

(1)2 * • • J the finite set of all elements of the form r^Ar^ '1

where 0 < A < e; 0 < t < 0 , jL # 1, j = 1, 2, . . , t. Now since

T is potent, it follows that for each positive integer x, elements

Vg; . . } v„ can be found so that each is a certain r^^ and the

product r^'Vg .. . v^ is different from zero. From the definition of

G. and a, it follows that if x > e , x > o, then the set v^, v^, . . , v^

necessarily contains the element r^ as well as elements different

from T]. Hence, by choosing an arbitrary integer y and fixing x so

that X > (E+a)(y+2), we havev^, . . , v % = fg^ . . . gyh, where f and

h are certain elements of T, while the g^ are elements of the set

Wg, . . 1 Since g^gg . . . gy f 0, it follows that the ring {w^,

Wg, • • , Wg} if. potent. Since either r^eR^ or r^eRg, it is true that

all the are either in R^ or in Rg which contradicts the assumption

that R^ as well as Rg is semi-nilpotent. 279

By Theorem 30, it follows, by induction, that the sum of any finite number of semi-nilpotent right ideals is again a semi-nilpotent right ideal. Using this fact we have

THEOREM V-31. The sum N of all semi-nilpotent right ideals is a semi-nilpotent two-sided ideal, which contains also all semi-nilpotent left ideals of the ring.

DEFINITION V-21. The sum N of all two-sided semi-nilpotent ideals of a ring R is called the Levitzki radical of the ring.

Remark

By Theorem V-31, the radical N also contains all one-sided semi- nilpotent ideals.

THEOREM V-32. If N is the radical of S, then the radical of ^ is zero.

Proof;

We show that if R 3 N is a semi-regular right ideal in S, then

^ is semi-regular in ^ . Let the elements r^, r2 , . . , of R generate the potent (not nilpotent) ring

T = {r^, tg, . . , r^}.

If E, is semi-nilpotent, then the ring N T = (T+N) K is nilpotent; that is, for a certain A we have that

T^ « Ô, where 0 is the zero of — , or Ç N. We denote byu., u_, . . , u the finite set of all products of the form r^ , r. , . . . r and put 1 ^2 U = {u^, u%, . . , u^}. 280

Since T, as well as for each X, is potent and since evidently

Ç U^T, we have that f 0 for each o. On the other hand by definition of U, we have that IT £ T^; hence U s N, which contradicts the fact that

N is semi-nilpotent. Thus the theorem is provedj that is, the radical

G of 4-S is zero. N The Levitzki radical is defined in terms of semi-nilpotent ideals and is actually the maximum ideal in the ring with this property. In rings with the minimal condition on left ideals, the Levitzki radical is identical to the classical radical. Levitzki also produced several results on a generalized radical which was defined to be the sum of all nil ideals of a ring. This radical is actually Baer's upper radical (see Definition V-26).

In the first section we explored Hopkin's work concerning rings with the minimum condition on left ideals and the classical radical.

In so doing, it was inevitable that semi-simple rings would be mentioned. However, we postponed a discussion of these rings and their association with the development of ideal theory until now for two reasons: (1) to discuss the contributions of Baer and Levitzki who employed the minimum condition, to a degree, in their development of radicals and (2) chronologically, the classical work on rings with the minimum condition on ideals, which includes a discussion of the semi-simple rings, was not published until 1944 which follows the dates of publication of Baer and Levitzki.

Some Aspects of Semi-simple Rings

Of all the rings that satisfy the minimum condition on left ideals, the semi-simple ones have the most profound results. This is 281 due to the fact that their structure can be completely characterized.

Simple rings are semi-simple since they have no two-sided ideals other than the zero ideal and themselves. Semi—simple rings are really special

Artinian rings. In their publication of 1944, Artin, Nesbitt and

Thrall[71, utilizing the classical radical of a ring, rad R, characterize a semi-simple ring in terms of simple ideals. We shall move in the direction of this characterization. Artin, Nesbitt and Thrall prove the following theorems, assuming that the rings satisfy the minimal condition on left ideals.

DEFINITION V-22. A ring R is said to be semi^-slmple if it has a zero radical and satisfies the minimal condition for left ideals.

THEOREM V-33. Any left ideal L in a semi-simple ring R contains an idempotent generator e; that is, L = Re.

Proof:

We first show tlîat any nonzero left ideal L in R contains an idempotent. Since R is semi-simple, it satisfies the minimal condition on left ideals. Hence it is sufficient to show that any minimal left ideal contains an idempotent. For any XcL, LX is a left ideal con­ tained in L, Since L is a minimal ideal, then either LX = 0 or LX = L,

If LX = 0 for every X in L, then L would be nilpotent, which is a con­ tradiction to the definition of a semi-simple ring. Hence it is true that for some X, LX = L. Thus L contains an element e such that eX = X and so eX-X = 0 which implies that (e®-e)X = 0. But the set of elements in L which annihilate X is a left ideal of R. But this ideal cannot be all of L since eX = X. But since L is minimal only OoL 282

annihilates A; that is, e^-e = 0 which implies that = e. Hence L

contains an idempotent e.

Secondly, we must show that in any nonzero left ideal L, there is

an idempotent e which is annihilated only by 0 in L; that is, Xe = 0,

XeL, implies that X = 0, With each idempotent e in L we associate the

left ideal consisting of all left annihilators of e in L. Let e be

an idempotent for which Mg is minimal. Such an e exists because of

the minimal condition on left ideals in R. Suppose that Mg 7^ 0 implies

that 6 ]^e = 0. Set e’ = e-ee^-e^. Then e'e = e = ee', e^e' = e^ = e'e%;

and so e ’e' = e; (e-ee^+ej^) = e'. Also since e'e = e. Mg 2 . But,

since e^e = 0 and e^e' 7^ 0 , Mg? is actually a proper subset of Mg.

This contradicts the minimal nature of Mg. Hence it is not true that

Mg 7^ 0 ; that is, Mg = 0 , or e is annihilated from the left by no non­

zero element of L.

Thirdly, we must show that the e chosen in the second part is a right unit for L and that L = Re. Let XeL. Then (X-Xe)e = 0. Hence

X-XeeMg, or X = Xe. This shows that e is a right unit for L and that

L = Le. Now Re 9 Le.= L since R ^ L. On the other hand. Re S L since e is an element of the left ideal L. Hence L = Re.

The next theorem characterizes unit elements in any two-sided

ideal of a semi-simple ring.

THEOREM V-34. Let a be any two-sided ideal in a semi-simple ring

R. Then a = eR. Further, e is uniquely determined by «.

Proof:

Since a is a right ideal, the set of all x in a for which ex = 0

is a right ideal g of R. Now eg = 0 and ge = g imply that 283

63 = (3e)B = 3(e3) = 0; that is, 3 is a nilpotent right ideal. Now

3+R3 is a two-sided nilpotent ideal, and so it must be zero, or else R

is not semi-simple. But we have that B+R3 = 0 if and only if 3=0.

(Or else, one can argue directly that 3=0, since a ring without

radical and with the minimal condition on left ideals can have neither

left nor right nilpotent ideals other than the zero ideal (0).) But

clearly

e(y-ey) = 0

for y in a, and this implies that y-ey is In 3 , that is, y-ey = 0

which says that y = ey. This shows that e is a two-sided unit element

of a. Now suppose that e' is also a unit element of a. We have that

e' “ e ’e = e.

Hence if any ring has a left unit element e ’and a right unit element e

they must coincide, that is, e' = e.

Corollary. Any semi-simple ring R has a two-sided unit element,

denoted usually by 1 .

He require two lemmas before we state and prove the main struc­

ture theorem for semi-simple rings.

LEMMA V-23. Any two-sided ideal a of a semi-simple ring R is

itself a semi-simple ring.

LEMMA V-24. Let ^2» ' * ; % distinct simple ideals of a

semi-simple ring R. Then the sum a = is direct.

THEOREM V-35. A semi-simple ring R contains only a finite number of simple ideals, and is the direct sum of them. Furthermore, any two-

sided Ideal of R is the direct sum of those simple ideals which it

contains. 284

Proof ;

Suppose we consider the set of ideals R(l-e) where e runs over the set of all finite sums of unit elements of simple ideals. But by the minimal condition in R, we can select e = 0 ^+6 2 + • • • so that

R(l-e) is minimal. Then

R = Re2 +Re2 + . . . Re^+R(l-e) is a direct decomposition of R. If R(l-e) f 0, then it contains a simple ideal Re^^^^« Set e' = e+e^^j^ and then the direct splitting

R(l-e) = R(l-e*)+Re^^^ by Theorem V-34 above. This shows that R(l-e') is properly contained in R(l-e). But this is a contradiction to the minimality of R(l-e).

But this contradiction arose from the assumption that R(l-e) was not zero. Hence R has only a finite number of simple ideals and is the direct sum of them. By Lemma V-23, any two-sided ideal of a semi­ simple ring R is itself semi-simple. Hence by this fact and the first part of our proof, any two-sided ideal of R is the direct sum of those simple ideals which it contains.

The theorem above expresses the decomposition of a semi-simple ring in terms of a finite number of simple ideals. We next consider the decomposition of such Ideals in terms of left Ideals.

THEOREM V-36. Let L be a left ideal in a semi-simple ring R, and suppose L is a sub-ideal of L. Then R contains an ideal L 2 such that

L is the direct sum of L]^ and L 2 ; that is,

L — L^ ® L 2 . 285

Proof :

Let L = Re and L^ = Rej^ where e and idempotents. This can be accomplished by Theorem V-34. We write any element A of L in the form

A = A 2+A 2, where Aj^ = Ae^ and A2 = A-Aej^ is a left annihilator of e. The set of elements in L which are annihilators of ej^ is a left ideal L 2 of R.

The fact that each AeL can be written as A = A%+x2 , where Aiehj^ and

A 2 EL2 , shows that

L — L2 +L2 .

But Ae^ = Aj^ shows that A = A^iA 2 is unique. Hence we have that

L ^ L^ ® L 2 .

Corollary. Any semi-simple ring can be written as the direct sum of a finite number of minimal left ideals.

Remark

Wedderburn[2l4], whom we discussed in Chapter II, proved the following: A ring R is semi-simple if and only if R is isomorphic to

^"1^ * ® ® ^n^^' where for i = 1, 2, . . . k, is a division ring, and is the ring of all n^xn^ matrices over

We shall close our discussion of semi-simple rings by at least discussing a few aspects of the general ring and its reduction to the semi-simple case. We shall illustrate these aspects by some results due to Artin, Nesbitt and Thrall[7].

Suppose R is a nonsemi-simple ring with the minimal condition on left ideals. As before, let rad R be the classical radical of R. Thus 286

THEOREM V-31. ^ is a semi-simple ring, rad R Proof; — R One must show that R = ÿ â 'd ~R zero radical and has the minimal condition on left ideals. Let [a] denote the residue class of the element a modulo (rad R). Then we call the mapping a + [a] the natural homomorphism of R into R. Now let M be a nilpotent left ideal of R and let M denote the set of elements of R which map into M under the natural homomorphism. M is a left ideal of R, M is nilpotent and therefore contains rad R. But this implies that M = rad R, so that M is the zero ideal of R. Now again by the application of the natural homomorphism we may see that any descending chain of ideals in R must terminate since the descending chain of ideals in R terminates.

Accordingly, the minimal condition holds in R and hence R = — -— is rad R semi-simple.

Gradually, the importance of radicals as a vehicle for studying rings, in particular non-commutative rings, .is beginning to emerge.

We have repeated from time to time the fact that there are several notions of a radical of a ring with corresponding notions of semi­ simplicity. Any theory of a radical attempts to prove a structure theory for semi-simple rings analogous to the case for rings with the minimal condition on left ideals.

Some Basic Properties of the Jacobson Radical

In his paper of 1945 Jacobson[95] gave another generalization of the classical radical of a general ring. Jacobson originally made use of elements of a ring which he called "quasi-regular" elements in characterizing his radical. 287

DEFINITION V-23. An element z of an arbitrary ring R is a right

quasi-regular if there exists an element z' in R such that

z+z’+zz' = 0.

Remarks

1. The element z ’ satisfying the above equation is called a

right quasi-inverse of z.

2. If R has an identity, then,z is right quasi-regular with

right inverse z' if and only if 1+z has an inverse 1+z'.

3. If z is an element of R, then the set of all elements of the

form x+zx, where x ranges over R, is a right ideal.

4. If z is a right quasi-regular with right quasi-inverse z,

then -z = z'+zz* e{ x+zx}.-

A right quasi-regular element z of a ring R has the following

alternate definition which is equivalent to Definition V-23:

DEFINITION V-24. An element z of a ring R is right quasi-regular

if the totality of elements

{x+zx} = R.

DEFINITION V-25. An ideal a (or a right ideal a) will be called

right quasi-regular if all the elements of a are right quasi-regular.

Remark

The sum of two right quasi-regular right ideals is quasi-regular.

Jacobson proved that

THEOREM V-38. If R is an arbitrary ring, then the intersection

of all the quasi-regular right ideals of R is a (right) quasi-regular

two-sided ideal. 288

Proof ;

Let J be the intersection of all quasi-regular ideals of R.

Since the right ideal generated by an element z is the totality of elements zi+za, where i is an integer and aeR, it is clear that J is

the totality of elements z such that zi+za is right quasi-regular for all integral i and all a in R. By the Remark to Definition V-25, J is a right ideal. We must show that J is a two-sided ideal. Let zeJ and beR. Then zbeJ and hence there exists an element w ’ such that

zb+w'+Czb)w’ = 0.

Now observe that

bz+(-bz-bw'z)+bz(-bz-bw’z) = b(w'+zb+zbw') z = 0

and so -(bz+bw'z) is a right quasi-inverse for bz. Similarly, if i is an integer and aeR, then

(bz)i+(bz)a - b(zi+za)

is right quasi-regular. Hence bzeR. Thus J is a (right) quasi- regular two-sided ideal.

In view of this theorem, Jacobson defines his radical as follows:

DEFINITION V-26. The radical of a ring R is the intersection J of all the quasi-regular right ideals of the ring. (Thus J is the

Jacobson radical of a ring.)

Remarks

1. If R is a ring with an identity, the connection between

quasi-regularity and regularity shows that J is the totality of

elements z such that 1+za has a right inverse for every a in R. 289

2. All of the preceding remarks for right ideals hold for left

ideals by similar arguments and analog]usly we can define left quasi­

regularity, left quasi-inverse, quasi-regular left ideals, and left

radicals.

3. An element is called quasi-regular if it is both right and

left quasi-regular.

A quasi-regular element can be characterized as follows:

THEOREM V-39. If z is quasi-regular, then any right (left) quasi­

inverse is a left (right) quasi-inverse and it is uniquely determined

and commutes with z.

Proof :

Let z' be a right quasi-inverse and z" a left quasi-inverse of an

element z. Then we observe that

z+z'+zz' = 0 and z"z"+z"z = 0.

Hence

z" = z"+(z+z'+zz')+z"(z+z'+zz') = z'+(z+z"+z"z)+(z+z*'+z*'z)z’ = z’.

Thus we see that any right (left) quasi-inverse is a left (right) quasi­

inverse . Now since

z+z'+zz' = 0 and z+z'+z'z = 0, then z+z'+zz* = 0 = z+z’+z'z = 0, which implies zz’ = z'z. This shows

that z' commutes with z. Thus the theorem is proved.

Remark

We call z' the quasi-inverse of z,

THEOREM V-40. The radical of a ring R is the intersection of all

the quasi-regular left ideals of R. 290

Proof ;

Suppose z e J. Then z has a right quasi-inverse z ' = -z-zz’. Now since J is an ideal z ’eJ. Hence z' has a right quasi-inverse. Now since z is a left quasi-inverse of z \ then by Theorem V-39, z' is quasi-regular and z is its quasi-Inverse, Hence z is quasi-regular.

Now since J is a left ideal, then J 5 J', where J' is the left radical.

Thus by symmetry J' S J.

Jacobson's radical shares many of the properties of the other radicals. In particular, we have

THEOREM V-41. Let R be an arbitrary ring and let R* be a ring with an identity containing R such that R * = R+(l). Then the radical

J(R) = J(R*)nR. If in addition R(1 (1) = (0) and (1) is isomorphic to the ring of integers then J(R) = J(R*),

Proof:

Suppose we assume that R* = R+(l). Now let J(R*) be the radical of R*, and let zeJ(R*)A R. Then z has a quasi-inverse z ' in R*. Since z' = -z-zz', z'eR and z is quasi-regular in R. Hence J(R*)riR s J(R).

Now since R* = R+(l), then any right ideal of R is a right ideal of R*.

Hence J(R) 2 J(R*) . Thus J(R) = J(R*) DR.

Now suppose that RH(1)= (0) and that (1) is isomorphic to the ring of integers. Then if z *eJ(R*) the coset z* of z* in the quotient ring ™ is in the radical of this ring. Since the radical of the ring of integers is 0, then z* = 0 and z*eR. Hence R 2 J(R*). Thus from the equation J(R) = J(R*)riR we obtain J(R) = J(R*). 291

Remarks

1. If R = J, we shall call R a radical ring.

2. If J = (0), then R is called a Jacobson semi-simple ring.

THEOREM V-42. If J is the radical of R, then R = & is semi-

simple.

Proof :

Let z be an element of the radical of R and let z be an element

in the coset z. Then there exists an element z ’ such that z+z'+zz' =

ueJ. Also there exists a u' such that u+u'+uu' = 0. Thus

0 = (z+z’+zz')+u'+(z+z'+zz')u' = z+(z’+u'+z'u')+z(z'+u'+z'u*).

Therefore z is right quasi-regular. Since the totality of elements z

in the cosets z of J(R) is an ideal, this totality is a quasi-regular

ideal. Hence z e J and z = 0. Therefore R is semi-simple.

In more modern times, as we shall see later in this chapter,

rings are studied via modules. Many important facts about a ring R can be learned if one knows something about the kinds of modules R will admit. In view of this modern trend, the Jacobson radical canbe

characterized in terms of modules.

DEFINITION V-27. Let R be an arbitrary ring. If there exist

irreducible R modules (see Appendix A), we let

J(R) = n A(M), M an irreducible r-module where A(M) is the annihilator of M in R (see Appendix A).

Remarks

1. If R has no irreducible R-modules, we set J(R) = R. 292

2. If J(R) = R» we say as usual that R is a radical ring.

3. Definition V-27 is equivalent to Definition V-26.

Some other properties of the Jacobson radical are:

THEOREM V-43, The Jacobson radical of a ring contains every nil right (left) ideal of the ring.

Corollary. If z is an element such that RzR 5 R, then zeJ.

THEOREM V-44. If M is a subring of J and if zeM, then for any positive integer h either z^“^M a z^M or z^ == 0 .

THEOREM V-45. If R is a ring with an identity whose lattice

(see Appendix A) of right (left) ideals is completely reducible, then

R is semi-simple.

Jacobson[95] applied his radical to several classes of rings to determine the general character of the radical of these rings. Some of the results are:

THEOREM V-46. If R is a ring that satisfies the minimal condi­ tion for left (right) ideals, then the Jacobson radical of R is nilpotent.

Proof :

Suppose that R is a ring for which the minimal condition for left (right) ideals holds. Let M be a two-sided ideal contained in J 9 and suppose also that M = M. Then if M f (0), there exists a minimal right ideal N of R with the properties (1) N S N and (2) Ntl (0).

Suppose that b is an element of N such thatbMf (0). Then (bM)M = bM

^ 0 and since bM 5 N, then we have that bM = N since N is minimal.

Since beN, there exists an element yeM such that by = b. But this 293 leads to b = 0 which is a contradiction since bM ^ 0. Thus M = (0).

Now we know that the positive powers of M are two-sided ideals and

M 2 -Î. . . . . Hence there exists an integer a such that

Then for M = we have that = M. Hence

M = = (0).

Thus J is nilpotent.

Let R be an arbitrary ring. We denote the ring of nxn matrices with elements in R by R^^,which is the usual convention.

LEMMA. V-25. Any matrix z = (Z^j) of R^ in which is right quasi-regular and the Z^j = 0 for i>j is right quasi-regular in R^.

THEOREM V-47. If R is an arbitrary ring and R^ is the ring of nxn matrices with elements in R, then the radical J(En) - where J is the Jacobson radical of E^,

Once a reasonable definition of aradical has been obtained, the next step is to characterize the rings which are semi-simple with respect to this radical. We need to talk about endomorphism rings.

Let N be a right ideal in R. Now if a is in R, then the right multiplication x +xa determined by a Induces an endomorphism "a in the difference group

G = R-N.

This mapping a' sends the coset x+N into xa+N. The set of all elements la is a subring R of the ring of endoraorphisms of G. The map a -s-a" is a homomorphism of R into R. It can be shown that the kernel of this homomorphism is a two-sided ideal N:R which we shall call the quotient of N relative to R. 294

DEFINITION V-28. A ring of endomorphisms is said to be

irreducible if the group G in which R acts is irreducible.

THEOREM V-48. Any irreducible ring of endomorphisms R is semi­

simple .

Proof:

Let zT be an element of the Jacobson radical J of R and let x f 0 be an arbitrary element of G. Now if R ^ (0) is irreducible, then the

set of all elements z in G such that zR = (0) is a subgroup of G; call

it S. Hence either S = (0) or S = G. Now since R f (0), S# G and so

S =(0). Thus if % ^ OeC, then xR (0), But since xR is a subgroup

invariant under R, then xR = G. So then if xz f 0, then (xY)R = G.

Hence there is an â in R such that xza = x. The element -aa has a

quasi-inverse z"'. Hence

X = x-(xza-xz'+xzaz') = x-xza+(x-xza)z' = 0 .

But this is a contradiction which shows that xz = 0 for all x. Thus

z = 0 and R is semi-simple.

Corollary. If N is a maximal right ideal, then (N:R) contains

the Jacobson radical J of R.

Jacobson[95] also defined his radical iq terms of maximal right

(left) ideals (see Corollary 2 to Theorem V-49 below.) Let R be any ring which is not a radical ring. Then R contains an element which is not right quasi-regular. Hence the right ideal {x+ax} does not contain a. If can be shown that if N is a right ideal that contains a and con­

tains the ideal {x+ax}, then N = R. The results pertinent to maximal

right ideals follow. 295

LEMMA V-26. If a is an element of R which is not right quasi­ regular, then the right ideal {x+ax} can be embedded in a maximal ideal.

THEOREM V-49. If R is a ring which contains maximal right ideals

and is the intersection of these maximal right ideals, then

^ ^ i - ^ and

RJ S ÇNi, where J is the Jacobson radical of R.

Corollary 1. If R is not a radical ring; that is, R if J and is the intersection of these maximal right ideals, then

^ and

RJ £ g Mi"

Corollary 2. If R is a ring with identity, then the Jacobson radical J of R is the intersection of the maximal right ideals of R.

Remarks

1. If R is a ring that contains maximal right ideals then

is a two-sided ideal.

2. The radical of a norraed ring (see Appendix C) is a closed ideal.

Some additional significant results on the radical as the inter­ section of maximal rights ideals are;

THEOREM V-50. Let R be an arbitrary ring that contains maximal right ideals. Then the radical of R is the intersection n(N:R), where

N ranges over the maximal right ideals of R. 296

Corollary. If R is not a radical ring, then

J = (N:R),

where N ranges over the maximal right ideals.

In the process of constructing a radical for a general ring, it

became necessary to find an analogue in the general ring for simplicity

in rings with the minimal condition on left ideals. The notion of

priraitivity was the required analogue.

DEFINITION V-29. A ring R is primitive if R contains a maximal

right ideal N whose quotient (N;P.) = (0) (see Appendix A).

THEOREM V-51. A necessary and sufficient condition that R be

primitive is that R be isomorphic to an irreducible ring of

endomorphisms.

Proof: = >

Suppose R is a primitive ring and let R be the ring of endo­

morphisms x+N + xa+N in G = y , which was discussed earlier. Then R is

irreducible and

R = R-(N:R) = R.

Thus any primitive ring is isomorphic to an irreducible ring of

endomorphisms.

Proof; < = =

Conversely suppose that R if (0) is an irreducible ring of endomor­

phisms acting in G. Suppose x is an element in G such that x f 0. Let

Njj be the set of all elements b of R such that xb = 0. It is true that

•Njj is a right ideal in R. It is also true that xR = G as has been

shown earlier. Thus N^cR, Suppose a is an element of R which is not

in Nj. and let ‘C be an arbitrary element of R. Then xlT 0. Hence 297

(xa)R « G and so there exists an element u such that xau = xc. Thus c’-âïTeNjj and R « N^+aR for any a not in Hence is maximal. If c / OeR, then there exists ay in G such that yc # 0. There exists an

a such that xa = y. This implies that xac f 0, Thus for any c f 0,

there is an a such that acjSN^^. On the other hand, if ce(N^^R), then

aceNx for all a. Hence c = 0 and (Nj,;R) = 0.

Let ^ family of rings. We denote by RR^ the set of all n-tuples {x^}, where x^cR^. We can define pointwise addition and multiplication which make RR^ into a ring, called the strong direct

sum (or direct product) of {^^n^neN’

DEFINITION V-30. A ring R is a subdirect sum of a family of rings there is an injective homomorphism

f :R->RRjj

such that for all meN

is a surjective homomorphism, where for each meN we define

by ° *m-

Remarks

1. Each is surjective.

2. Each is called the mth projection.

The subdirect sum is the analogue of the structure theorem for the

classical radical. In view of this we have

THEOREM V-52. A ring R is semi-simple if and only if it is a

subdirect sum of primitive rings. 298

It is interesting to observe that for rings with the minimal condition on left ideals that the Baer, classical, Jacobson,and

Levitzki radicals all coincide.

Prime Ideals in Noncommutative Rings

McCoy, as we shall see, is not only responsible for a notion of a radical but also for some results on prime ideals in noncommutative rings. The concept of a prime ideal, which we have encountered

several times in our development of ideal theory, is one of the most

fundamental concepts in all ideal theory. In Chapter III We discussed

this concept in conjunction with the results of Lasker and van der

Waerden; in Chapter IV, we, in our summary of the works of Noether,

Krull and van der Waerden, saw the versatility of prime ideals again

and finally, in the first part of this chapter we have seen extensive use made of this concept. In all of these cases, however, the authors were concerned with prime ideals in commutative rings. McCoy[152] decided to find out what role these ideals play in the study of non­

commutative rings. He proved the following:

THEOREM V-53. If p is an ideal in the arbitrary ring R with

identity, then the following conditions are equivalent:

1. If a and g are ideals in R such that ag = 0(p), then a = 0(p)

or 3 E 0 (p).

2. If (a), (b) are principal ideals in R such that (a)(b) = 0(p),

then a = 0 (p). or b = 0 (p).

3. If aRb = 0(p), then a = o(p) or b = 0 (p ).

4. If I^, Ig are right ideals in R such that 1 ^ 1 2 = 0(p), then

I^ = 0 (p) or Ig = 0 (p). 299

5, If are left ideals in R such that Lj^L2 = 0(p), then

- 0 (p) or L2 - 0 (p).

Proof :

1. = > 2. Clearly 1. = » • 2. by setting a = (a) and g = (b).

2. ===> 3, Suppose that if (a) and (b) are ideals in R such

that (a)(b) = 0(p), then a = 0(p) or b 5 0(p). Also suppose that aRb

= 0(p). Now aRb = 0(p) implies that RaRbR = 0(p), and thus (a)^(b)^ —

RaRbR = 0(p) implies that a = 0(p) or b 5 0(p). Hence 2. = = > 3.

3. = > 4. Suppose that aRb = 0(p) implies that a = 0(p) or

b = 0 (p) and suppose that I2 are right ideals such that I^l2 = 0 (p)

with ^ 0(p). Let a^ be an element of not in p. Then for every

element 8 2 of Ig we have a^Ra2 ~ I1 I2 ~ 0(p)' Hence by 3. we have

^ 2 = 0(p). Thus I2 = 0(p), and we have shown that 3. = > 4.

3. = > 5. Analogous to 3. = > 4.

4. = > 1. Clearly 4. = = > 1. by setting = a and I2 = g*

DEFINITION V-31. An ideal p with any one (and therefore all) of

the properties stated in Theorem V-53 is a prime ideal.

LEMMA V-27. If p is a prime ideal in R and a an element of R

such that RaR = 0(p), then a = 0(p).

LEMMA V-28. If g is an ideal in R and p a prime ideal in R,

then g n p is a prime ideal in the ring g.

DEFINITION V-32. A set M of elements of R is an m-system if and only if CeM, dsM imply that there exists an element x of R such that

cxdeM. 300

Remarks

1. The empty set Is an m-system.

2. By Part 3 of Theorem V-53, an Ideal p in R is a prime ideal if and only if its complement C(p) in R is an m-system.

3. An m-system is a generalization of a multiplicative system.

(The system used in the construction of quotient rings.)

McCoy utilized much of the work of Krull[109, 113] in his dis­ cussion of the radical of an ideal. We have previously focused on the radical of the ring itself. We nowlook at the radical of an ideal of a ring along with the radical of the ring.

DEFINITION V-33, The radical r of an ideal a in a ring R con­ sists of those elements x of R with the property that every m-system which contains x contains an element of a.

Remarks

1. An ideal a is contained in its radical r.

2. a and r are contained in the same prime Ideals.

McCoy defined his radical of a ring in terms of the radical of an ideal.

DEFINITION V-34. The radical N of the ring R is the radical of the zero ideal in R.

Remarks

1. N is a nil ideal.

2. N is the intersection of all the prime ideals in R.

3. aR 5 N if and only if a is an element of N.

4. Every element b which generates a nilpotent ideal (right, left or two-sided) is in N. 301

DEFINITION V-35. A prime ideal p is a minimal prime ideal belonging to the ideal a if and only if « E p and there exists no prime ideal p ’ such that a E p ' E p.

LEMMA V-29. Let a be an ideal in R and M an m-system which does not intersect a. Then M is contained in an m-system which is maximal in the class of m-systems which do not intersect a.

LEMMA V-30. Let M be an m-system in R and.a an ideal which does not meet M. Then a is contained in an ideal p* which is maximal in the class of ideals which do not intersect M. The ideal p* is necessarily a prime ideal.

LEMMA V-31. A set p of elements of the ring R is a minimal prime ideal belonging to a if and only if C(p) is maximal in the class of m-systems which do not meet a.

THEOREM V-54. The McCoy radical r of an ideal a is the inter­ section of all minimal prime ideals belonging to a.

Proof:

Suppose r is the radical of a ; then r is contained in the same prime ideals as a. This implies that r is contained in the inter­ section of all the minimal prime ideals belonging to a. Let a be an element of R not in r. Thus, by the definition of r, there exists an m-system M which contains a but does not intersect a. By Lemma 31,

C(M') is a minimal prime ideal belonging to a and clearly C(M') does not contain a. Hence a cannot be in the intersection of all minimal prime ideals belonging to a and hence r is the intersection of all the minimal prime ideals belonging to

Corollary. The McCoy radical of an ideal is an ideal. 302

THEOREM V-55. If R is a ring which satisfies the minimal con­ dition for left (right) ideals, then N coincides with the classical radical of R.

Proof;

Suppose aeN; then (a) is a nil ideal. Since R satisfies the minimal condition for left (right) ideals, then (a) is a nilpotent ideal. But as is indicated in remark 4 to Definition V-34, N contains all elements which generate nilpotent ideals. Hence N consists precisely of the elements which generate nilpotent ideals. This is one of the familiar characterizations of the classical radical. Hence the theorem is proved,

THEOREM V-56. If g is an ideal in R, then the radical of the ring B is g O N.

Proof;

If N ’ is the radical of the ring g , by Lemma 28 N' 5 gHN. On the other hand, if beg A N, then every m-system in R which contains b contains 0. Thus in particular every m-system in g which contains b contains 0. This implies that bgN’ and thus g A N - N'. Hence the radical of g is g AN.

THEOREM V-57. If N is the McCoy radical of R, then ^ has zero radical.

Proof:

Suppose a is an element of N, where N is the radical of — . N — n _ Hence a is contained in all prime ideals o f — , If a f 0, then N a i 0(N) and hence a is not contained in some prime ideal p in R. Since 303 p ® N by one of Noether's ring isomorphism theorems (Theorem IV-20), we have

R .

This implies that ^ is a prime ideal in & Further, it is true that — " " • N does not contain "a since a ^ 0(p). This is a contradiction since a is R — R contained in all prime ideals of — . Thus a = 0. Hence — has zero ^ N N ideal.

Remarks

1. ^ contains no nonzero nilpotent ideals (right, left or two- sided) .

2. N is a radical ideal as defined by Baer. (See the section titled Some Fundamentals of Radical Ideals, above.)

The constituent building blocks for the structure of rings with zero McCoy radical are prime rings.

DEFINITION V-36. A ring R is prime if and only if (0) is a prime ideal in R.

Remarks

1. A commutative prime ring is an integral domain.

2. A simple ring S (with ^ 0) is a prime ring.

3. A primitive ring is a prime ring,

4. An ideal in a prime ring is a prime ring.

In Chapter II we gave a discussion of Wedderburn, who made significant contirubtions to modern algebra and ring theory.

Wedderburn*s work has been modified and extended by Artin, whom we 304 discussed in Chapters IV and V. For this reason, many of Artin's theorems are referred to as the Artin-Wedderburn theorems. McCoy, us- ing the fact if p is a prime ideal in an arbitrary ring R, then — is a prime ring and conversely, and the fact that N is the intersection of all prime ideals inR, proved a theorem which is an analogue of an

Artin-Wedderburn theorem. The theorem follows below.

THEOREM V-58. A necessary and sufficient condition that a ring be isomorphic to a subdirect sum of prime rings is that it has zero radical. „

Another result of McCoy is

THEOREM V-59. A prime ring that contains minimal right ideals is a primitive ring.

Suppose T is a ring with unit element and that T^ is the ring of all matrices of order n with elements in T. McCoy was able to deduce several results concerning prime rings in the class of matrix rings.

THEOREM V-60. If T is a ring with unit element, then T^ is a prime ring if and only if T is a prime ring.

Proof : = = >

Suppose we let (e^^)denote the matrix with unit element in the i-th row and j-th column with zeros elsewhere. If T is not prime, then

T^ is not prime. If T is not a prime ring, then there exist nonzero elements a, b of T such that

aTb = 0.

This implies that

(aei^)T_^(be^P = 0 305 with ae-, and be,, nonzero elements of T . This shows that T is not a 11 11 n n prime ring. Thus we have that is a prime ring Implies that T is a prime ring.

Proof : < = =

Suppose that T^ is not a prime ring. Hence there exist nonzero matrices (a^^), (b^^) in such that

Suppose we assume that a f 0, b f 0. Now for every x in T, we have PQ 3TS that

But the coefficient of e must be zero: that is, ps ’

and T is not a prime ring. Hence T is a prime ring implies that T^ is a prime ring.

The above theorem (Theorem V-60) is used to prove a theorem which yields a result which is convenient to have.

THEOREM V-61. If N is the McCoy radical of an arbitrary ring R, the radical of the complete matrix ring R^ is N^,

Proof :

Suppose R is an arbitrary ring with unit element; then there is a one-to-one correspondence

M M n between the ideals in R and the ideals in R . Also it is true that

■s>.4n and by the preceding theorem (Theorem V-60), M^ is a prime ideal in R^ 306 if and only if M is a prime ideal in R. Thus if N is the radical of R and the p^ are the prime ideals in R, then the

radical of R = A (p.) = (f^p.) = N . n i n i n n Now if R does not have a unit element, it can be shown that R can be imbedded in a ring S with a unit element in such a way that R is an ideal of S. If the radical of R is N and the radical of S is N*, then by Theorem V-56

N = RAN'.

Hence by the result above the radical of S is N and since R is an n n n ideal in S^, again Theorem V-56 shows that

radical of R = N'f'lR = (N'f^R) = N . n n n n n Thus the radical of the complex matrix ring R^ is N^.

McCoy's results extended the concept of a prime ideal to rings in general and gave still another structure theorem for noncommutative rings.

In his paper of 1952, Murdoch[159] uncovered some more results on noncommutative rings using the methods of McCoy[152], explored above, in his attempts to extend the Krull-Noether theory of commutative rings to noncommutative rings. His work differed from McCoy's work in that he did not make use of the results of Fitting[51] and Krull[113] as

McCoy did. Upper and lower isolated component ideals were the main vehicles of Murdoch's analysis.

DEFINITION V-37. If a is any ideal in R and M is an m-system which does not intersect a, the right upper M-component of a is defined to be the set of all elements x of R having the property that every right M-n-system which contains x intersects a. (See Appendix A.) 307

Remark

The right upper M-component is denoted by p(a, M).

DEFINITION V-38. If a is any ideal in R and M any m-system which

does not intersect a, the right lower isolated component of a corre-

^onding to M, is defined to be the set of all elements x of R such that xRm - a some element m of M.

Remark

The right lower M-component of a is denoted by &(o, M ) .

In view of these definitions, we have these main results.

THEOREM V-62. The right upper isolated M-component p(a, M) of a

is an ideal. Its complement in R is the uniquely determined maximal

right M-n-system which does not intersect a and y (a, M) itself is the

greatest common divisor of all ideals containing a which are in M and

right prime to a.

THEOREM V-63. If a is an ideal and M an m-system which does not

intersect a, then

y (a, M) 2 &(«, M) î a.

THEOREM V-64. (a) y[y(a, M), M] = y(a, M ) ,

(b) a[y(a, M)„ M] = y(a, M ) ,

(c) y[a(a, M), M] = y(o, M ) .

THEOREM V-65. If a is an ideal in a commutative ring R, and M an m-system which does not meet a, the set a(M) of all elements x of R for which xmea for some element m of M is an ideal.

Remark

The ideal a(M) defined in Theorem V-65 is called the isolated

M-component of a. 308

THEOREM V-66. If a is any ideal in a commutative ring R, and M

is an m-system which does not intersect a, then

y(a, M) = £(a, M) = a(M).

Curtis[28], in his paper of 1952, made use of McCoy’s results to present some contributions to the structure theory of noncommutative

ideal lattices as developed by Krull[113]. In addition to McCoy’s

findings, Curtis also drew from the following mathematicians whose works we have already discussed; Baer[10], Fitting[51], Fuchs[54],

Jacobson[95], Krull[106, 107, 108, 113], Letitzki[139], and Noether

[163], Curtis made use of primal ideals in his study.

DEFINITION V-39. a is right primal if the intersection P of the

ideals not right prime to a is again not prime to a.

Remark

If a is primal, then P is called the adjoint ideal of a.

The main result of Curtis’ investigations is the following:

THEOREM Vr67. Every ideal in R, where R is a noncommutative ring with identity 1, is the intersection of its primal divisors. If R

satisfies the ACC for ideals, then every ideal in R is an intersection of a finite number of primal ideals.

Previously in Chapters III and IV, the concept of a primary ideal was discussed along with related concepts concerning prime ideals.

There are many occasions when it is desirable to discuss these related

or associated prime Ideals without involving the primary ideals. Cer­

tainly this is exactly what Krull[107, 113] did for commutative rings

and later for noncommutative rings. Curtis utilized Krull’s methods in 309 his analysis of associated primes and considered only rings which satisfy the ACC for ideals.

In addition to his work on associated prime ideals, Curtis proved several results concerning primary ideals in rings which satisfy the

ACC for ideals. He, along with McCoy and Murdoch, was instrumental in determining the direction of the "Krull-Noether Theory" in noncommuta­ tive rings. We shall conclude this train of thought hy investigating a paper published by Barnes in 1956.

Wilfred Barnes was a student of Murdoch whose contributions to ideal theory were discussed in a previous section of this chapter.

Murdoch, as we recall, was concerned with extending some of the classi­ cal "Noether Theory" to noncommutative rings. Barnes continued this line of investigation under the direction of Murdoch, having become attracted to this aspect of noncommutative ring theory.

Barnes[13] consistently made use of the concepts on primal ideals which were uncovered by Curtis[28], Fuchs[54],and Murdoch. In particular, by considering ideals which have reduced representations by ideals with prime adjoints, Barnes extended much of Fuchs' results on commutative rings to noncommutative ones, Barnes discovered that rings which satisfy the ACC for ideals were the desirable candidates for these extensions. Barnes also used the definition of an m-system given earlier in our discussion by McCoy[152] and the definition of a right

M-n-system given by Murdoch in his investigations of the isolated components of an ideal. Bames, however, did make one exception. If

M is the empty set, then any set in R is a right M-n-system. Previously, . 310

In the case of Murdoch, if M is the empty set, then the only M-n-system

is the set M itself.

We conclude our discussion of the development of ideals in non­

commutative rings. We cannot overemphasize the fact that this chapter

does not represent all of the contributions to the development of ideal

theory. We have only cited some of the major milestones in the

development of this versatile algebraic structure in modern times up

through the decade of the 1960's. The volumes of research in this

field of mathematics during the decade of the 1960's alone is voluminous.

Now that we have traced the birth of ideals in algebraic number fields

and polynomial rings, through their period of adolescence during the

decade of the 1920's to their maturity in modern times, we naturally

question their relative position with respect to disciplines other

than algebra. In other words, what are some of the "non-algebraic"

applications of ideals? CHAPTER VI

SOME APPLICATIONS OF IDEAL THEORY

1. Some Applications of Ideals In Analysis and Topology

Applications of Boolean Rings to Point-set Topology

Though ideal theory was still in a phase of development during the decades of the 1930's and 1940’s, this period witnessed several applications of rings and ideals to some of the basic theories of analysis and topology. Most of these applications, however, were either based on or inspired by the now classic paper by Marshall Stone.

Stone[191], however, based this paper on his earlier paper[192] in which he discussed the theory of representations for Boolean algebras.

Since Stone’s former paper involved applications of Boolean rings to point-set topology, we find it necessary to cite some of the basic properties of these Boolean rings.

DEFINITION VI-1. A ring in which every element is idempotent, that is, satisfying the law

a^ = a, is called a Boolean ring.

The next theorem completely characterized these rings.

THEOREM VI-1. Let R be a Boolean ring, then

1. R is commutative.

311 312

2. R contains divisors of zero if it contains more than two elements.

3. Every Boolean ring R can be embedded in a Boolean ring R’ which possesses a unit element in such a manner that R' is unique in the following sense: If S is a Boolean ring with unity, containing R, then

S contains also a Boolean ring R" isomorphic toR* and containing R.

4. A finite Boolean ring necessarily possesses a unit and has a cardinality which is a power of 2.

Stone[192] was successful in establishing the basic relations between Boolean algebras and Boolean rings. Boolean algebras, some­ times referred to as "algebras of logic," maybe defined by the following postulates due to Huntington[91], a leader of the axiomatic approach in mathematics.

DEFINITION VI-2. A Boolean algebra K is a class of elements a, b, c, . . . satisfying the following postulates, where v is a binary operation which corresponds to logical addition and to the for­ mation of the union of classes and '' is a unary operation which corresponds to logical negation and to the formation of the complement for classes.

POSTULATE 4.1 If a and b are in the class K, then avb Is in the class K.

POSTULATE 4.2 If a is in the class K, then o' is in the class K.

POSTULATE 4,3 avb = bva.

POSTULATE 4.4 (avb)vc = av(bvc).

POSTULATE 4.5 ava = a.

POSTULATE 4.6 (a^vb^)'v(a^vb)^ = a. 313

Remark

Huntington[91] later discovered that Postulate 4.5 is redundant and therefore can be omitted from the set of postulates.

Some of the basic relations between Boolean algebras and Boolean rings are given in the results which we shall state without proofs.

THEOREM VI-2. If R is a Boolean ring with unit e, the introduc­

tion of a binary operation v and a unary operation " through the

equations

1. avb = a+b+ab

2. a' = a+e

converts R into an algebraic system B in which

a. avb = bva

b. av(bvc) = (avb)VC

c. (a^vb'') ^v(a^vb) = a,

the old operations being expressed in terras of the new through the

equations

3. a+b = ab"va''b - (a‘'vb'"')''v(a''''vb'')'',

4. ab = (a’'vb'')''.

Remarks

1. If B is an algebraic system satisfying laws a., b,, and c .,

then B is a Boolean algebra, and the introduction of new operations

through the laws (equations) 3. and 4. converts B into a Boolean ring

R with unit e = ava'” and zero 0 = e" = (ava")", the old operations

being expressed in terms of the new through the equations 1. and 2,

above. 314

2. The theorem above identifies Boolean rings with unit and

Boolean algebras, as characterized by Huntington's postulates.

3. In view of this theorem, a Boolean ring without unit can be regarded as imbedded in one which has a unit.

In addition to Huntington's characterization of Boolean algebras, there are several others. Most of these are found in Stone's[192] paper.

Stone introduced an abstract relation which corresponds to the relation of class-inclusions.

DEFINITION VI-3. In a Boolean ring R, the element a is said to be less than or to be contained in the element b, symbolized a< b, and the element b is said to be greater than or to contain the element a, in symbols b > a, whenever any of the equivalent relations

1. ab = a; 2. avb = b; 3. ab'=0; 4. a"vb - e is satisfied, relations 3. and 4. being significant if and only if R has a unit e.

Remark

The equivalence of relations 1., 2., 3., and 4. is more evident if they are written in terms of ring operations

1 *. ab = a ; 2'. a+b+ab = b ; 3'. ab+a = 0;

4'. (a+e)=b+( ab+a) = e.

It is inevitable that Stone's investigations would lead to results concerning subrings of rings, subclasses of classes, subsystems of systems and how they are interrelated. The next theorem supplies some detailed information concerning the subring generated by a given subclass. 315

THEOREM VI-3, In order that the subring a(c), generated by a non-empty subclass c of a Boolean ring R have a unit, it is necessary and sufficient that c contain elements a^, a^, . . , a^ such that

b < a,va„v . . . va_ 1 2 n for every element b in c. When this condition is satisfied, the element

a = a,va V . . . va 1 2 n is the unit of a(c) and a(c) is the class of all elements which can be constructed as polynomials in terms of the elements b and a+b, where b is in c , and of the operations v and • alone. In particular, if R has a unit e and c contains e, then a(c) is the class of all elements which can be constructed - as polynomials in terms of elements b and b ' = b+e, when b is in c and amenable to the operations v and • alone.

A discussion of subrings naturally leads to a discussion of ideals. Thus analogous to the last theorem (Theorem VI-3), we have

THEOREM VI-4. If c is an arbitrary non-empty subclass of a

Boolean ring R and if a(c) is the class of all elements a such that

a < aj^va2 va^v . . . va^ for appropriate elements a^, ag, . . j a^ in c, then a(c) is an ideal and every ideal containing c contains a(c). (The ideal a(c) may be characterized alternatively as the class of all elements a such that a = a^^b^^va^bgV . . . va^b^, where a^, a^ . . j a^ are in c and b^, b^,

. . , b^ are in R.) If c is the union of a set D of ideals a, then a(c) is the class of all elements a such that

a *= aj^va^v . . . va^, where each a^ is in and is in D for k = 1, 2, 3. 316

As a discussion of subrings of a ring would be incomplete without a discussion of ideals, a discussion of ideals of a ring would be in­ complete without a discussion of the principal ideals of a ring.

DEFINITION VI-4. In a Boolean ring R, an ideal a is said to be

1. principal if a = a(a) for some element a,

2. semiprincipal if a = u(a) or a = «'(a) for some element a,

3. simple if ctvci’ = e, where ea = a and ave = e;

4. normal if a = a".

Remarks (notations)

1. The classes of principal,-semiprincipal, simple, and normal ideals are denoted by the letters P, P*, G,and R respectively.

2. In general, the class of all ideals will be denoted by the letter j .

3. The coincident classes of divisorless, prime, and primary ideals in a Boolean ring R will be denoted by H.

The ideals in a commutative ring can be classified in terms of the relation of inclusion. The relation of inclusion establishes three important types of ideals; divisorless ideals, prime ideals, and primary ideals. These terms have been encountered in many of our earlier discussions, however, in another context. In view of the previous definitions, the divisorless ideal is the only concept which requires that its definition be stated at this time.

DEFINITION VI-5. An ideal a is said to be divisorless if it is distinct from the ideal s consisting of all elements of the ring and has no ideal divisors other than a and e. 317

Stone[192] pointed the fact that in the case of Boolean rings, the arithmetic theory of prime ideals is equivalent to the theory of representations. In view of this fact, prime ideals play an important role in the theory of Boolean rings. -

THEOREM VI-5, The Boolean ring R' of Theorem VI-1 contains R as a prime ideal when R has no unit; and the system P is a prime ideal in

P* when P ^ P*.

THEOREM VI-6. In a Boolean ring R, the classes H, R', and p*, the coincident classes of divisorless, prime and primary ideals in a

Boolean ring, satisfy the inclusion relation

HRCP*.

More precisely, an ideal p is both prime and normal if and only if p = a'(a), where a is an atomic element; and a prime ideal p fails to be normal if and only if

p' = (0).

THEOREM VI-7. If p is a prime ideal in a Boolean ring R and a

is an arbitrary ideal, then at least one of the relations acp, a'crp

is valid; if a is simple, then only one is valid.

In his constructions of algebra of classes. Stone[192] describes

a canonical form for concrete Boolean rings. In so doing, he shows how any algebra of classes can be reduced to canonical form.

DEFINITION VI-6. If R is a Boolean ring with elements a, b, c

. . . which are subclasses of a fixed class

E = E(R) with elements a, B, y, . . . » then R is said to be a reduced algebra 318 of classes when it has the following property: Every element a in E is

contained in some element of R, and is the only element of E common to

all the elements of R containing it.

DEFINITION VI-7. An algebra A of subclasses of a class E is said

to be perfect if

E(a) = E(B)

implies that

a = 3, where E(a) and E(3) are the unions of all those subclasses of E which

belong to the ideals a and 3 respectively.

The next theorem is a result which characterizes perfect

algebras of classes.

THEOREM VI-8. In order that an algebra A of subclasses of a

class E be perfect it is necessary and sufficient that

1. Every ideal other than e be the product of all its prime

ideal divisors.

2. E'(p) be a one-element class whenever p is a prime ideal.

Remark

Condition 1. of Theorem VI-8 is known as the "Fundamental

Proposition of Ideal Arithmetic."

The next theorems are necessary in order to establish the

Fundamental Proposition of Ideal Arithmetic for rings and perfect

representations.

THEOREM VI-9. In a Boolean ring R, every ideal other than e is

the product of all its prime ideal divisors. 319

THEOREM VI-10. Let R be a Boolean ring, a an arbitrary ideal in

R, H the class of all prime ideals in R, the algebraic system of all

Ideals in R under the operations of unrestricted addition and finite multiplication, H(a) the class of all prime ideals which are not divisors of a, and ^(A) the algebraic system with the classes H(a) as elements and with the operations of forming unrestricted unions and finite intersections. Then the correspondence

a H(ot) determines an isomorphism

(A) in accordance with the relations;

1. H(a) = H(3) if and only if

a = 3.

2. If B is any non-empty class of ideals, then

3. H(ec6) = H(t»)H(S).

Let H(a) denote the class H(n(a)) corresponding to the principal ideal a(a) and let R'(R) be the algebraic system with the classes H(a) as elements and with the operations of forming finite unions, symmetric differences,and finite intersections. Then R'(R) is a concrete Boolean ring or algebra of classes, isomorphic to R by virtue of the correspondence

a -»• H(a) in accordance with the relations 320

4. H(a) = H(b)

if and only if a = b; 5. H(a+b) = H(a)AH(b), the unique solution of xvH(a)H(b) = H(a)vH(b) and x(H(a)H(b)) = 0; 6. H(avb) = H(a) C H(b);

7. H(ab) = H(a)H(b).

The system R'(R) is a perfect reduced algebra of classes.

DEFINITION VI-8. The algebra of classes R ’(R) with a Boolean ring

R defined in Theorem VI-10 is called perfect representation of R .

THEOREM VI-11. Let R, H, H(a), I(R), H ( a ) , and R'(R) have the meanings specified in Theorem VI-10. Let J be an arbitrary subclass of

H; a(J’) the ideal consisting of all elements a such that H(a)cJ';

I(R, J) the algebraic system of all classes JH(a) under the operations

of forming unrestricted unions and finite intersections; and R'(R, J)

the algebraic system of all classes JH(a) under the operations of form­

ing finite unions, symmetric differences and finite intersections.

Then the correspondence

JH(a)

determines the homomorphisms

I(R) I(R, J),

I(R, a(J')) -+ i ( R , J ) ,

the latter of which is an isomorphism if and only if

H(a(J')) = J»

or equivalently,

J = H'(a(J')). 321

Similarly, the correspondence

H(a) = JH(a) determines a homomorphism

R ’(R) -> R'(R, J) and an isomorphism

R-(R, J)

The algebra of classes R'(R, J) is perfect if and only if H(a(J’))

H(a(J')) = J'; and when this condition is satisfied R' (R, J) is equivalent to R' (— •

If 3 is an arbitrary ideal, then we have, in particular, the result that R ’ (•§) is equivalent to R ’(R, H ’(3)). The ideal a(J’) is equal to p e when J is empty and equal to to the product of the prime ideals in J otherwise.

THEOREM VI-12. If R' is an algebra of classes homomorphic to a

Boolean ring R and if 3 is the ideal in R determined by the homomorphism

R + R', then there exists a class J of prime ideals in R related to 3 through the equation

a(J') = 3 such that R' is equivalent to R'(R, J). In order that R' be perfect, it is necessary and sufficient that

J = H'(3).

The only perfect algebras of classes isomorphic to R are those equiva­ lent to R'(R). 322

The preceding definitions, theorems, .and remarks representing only a minute portion of the theory of Boolean rings. Boolean algebras,and their representations. We have pointed out those highlights which are essential to a discussion of applications of this theory to topology.

In particular, as has been our general theme, we are concerned with the role that ideals play in such theory. Earlier, in our preliminary dis­ cussion, we discussed the construction of the perfect representation of

Boolean ring. We shall see that it is natural to impose upon H, the class of all prime ideals in a Boolean ring R, a neighborhood topology based upon the special subclasses H(a), where a is an ideal in R, or upon the more restricted subclasses

H(a) = H(a(a)), where a is an arbitrary element in R. Stone[191] outlined the con­ sequences of this procedure in the following theorem.

THEOREM VI-13. Let R be a Boolean ring, a an arbitrary ideal in

R, H the class of all prime ideals in R, H(a) the class of all prime ideals which are not divisors of a, and H(a) the class H(a(a)) corres­ ponding to the principal idea a = a(a). The topologies imposed upon H through the introduction of neighborhood systems 1, and 2., where

1. Each H(a) is assigned as a neighborhood of every element which it contains;

2. Each H(a) is assigned as a neighborhood of every element which it contains; are equivalent. Under them, H is a topological space with the properties; 323

1. H is a totally disconnected locally compact Hausdorff space;

2. The classes H(a) are characterized as the open sets in H;

3. The classes H(a) are characterized as the compact open sets in H.

The character of H is equal to the cardinal number of R whenever R is infinite. The space H is compact if and only if R has a unit e.

Stone[191] also proved the converse of Theorem VI-13.

THEOREM VI-14. If G is a totally disconnected locally compact

Hausdorff space, then the compact open subsets of G have a Boolean ring structure, call it R; and if the class H of prime Ideals in R is topologized as in Theorem VI-13, then H and G are topologically equiva­ lent (see Appendix F).

Theorems VI-13 and VI-14 establish a correspondence between

Boolean rings and certain topological spaces. In view of this corres­ pondence, we have

DEFINITION VI-9. A totally disconnected locally compact

Hausdorff space is called a Boolean space.

The next theorem reveals some of the topological facts pertinent to Boolean spaces.

THEOREM VI-15. 1. The closed subsets and the open subsets of a Boolean space are Boolean spaces. 2. A continuous image of a compact Boolean space is a compact topological space.

Stone further collected, in a new form, the results of Theorems

VI-13 and VI-14 of the introductory material above as well as some others in order to prove some results which develop the consequences of the relations between rings and topology in much greater detail. 324

Since the proofs of these results are rather long and depend heavily on the previously stated Theorems VI-1, VI-2, VI-10, VI-11, and VI-12, we shall not repeat them here. These results follow.

THEOREM VI-16. The algebraic theory of Boolean rings is mathe­ matically equivalent to the topological theory of Boolean spaces by virtue of the following relations:

1. Every Boolean ring has a representative Boolean space; every

Boolean space is the representative of some Boolean ring; and two

Boolean rings are isomorphic if and only if their representatives are topologically equivalent ;

2. The group of automorphisms of a Boolean ring is isomorphic to the topological group (see Appendix F) of an arbitrary representative of the ring;

3. The representatives of Boolean rings which are isomorphic to the various ideals in a Boolean ring are characterized topologically as open subsets of an arbitrary representative of R; in particular, H(a) is a representative of the ideal a in R;

4. The representatives of the homomorphisms of a Boolean ring R are characterized topologically as the closed subsets of an arbitrary representative of R; in particular, H'(a) is a representative of the quotient ring — ;

5. The representatives of Boolean rings with unit are character­ ized by the property of compactness.

Earlier in our discussion, we defined principal, semiprincipal, and simple ideals and proceeded to discuss the properties of classes of 325

Ideals. It is interesting to observe the topological characterization of these ideals. We shall find that the perfect representation H(R) is a most convenient way of stating the results which follow:

THEOREM VI-17. If a is an ideal in a Boolean ring R, then H(a') coincides with the exterior of H(a). Hence the sets H(a) corresponding to the ideals in the respective classes J, R, G, P*, P of Definition

VI-4, discussed in the preliminary results, are characterized topologi­ cally as follows:

1. The sets H(a) corresponding to arbitrary ideals are the open sets in H.

2. The sets H(a) corresponding to the normal ideals are the regular open sets in H.

3. The sets H(a) corresponding to simple ideals are the open and closed sets in H, or equivalently, the sets in H with empty boundaries.

4. The sets H(a) corresponding to the semiprincipal ideals are the open sets in H which are compact or are complements of compact open sets.

5. The sets H(a) corresponding to principal ideals are the compact open sets in H.

The above theorem, via Theorems VI-6, VI-8 and VI-10 discussed earlier, gives rise to some results which characterize the sets cor­ responding to prime ideals.

THEOREM VI-18. The set H(p) corresponding to an arbitrary prime ideal p in a Boolean ring R is the open set H-{p}; the ideal p is 326 normal, and hence semi-principal, if and only if p is an Isolated point in the Boolean space H = H(R).

THEOREM VI-19. The representatives of Boolean rings R' with the properties

1. R' is isomorphic to a subring B of the ring G of all simple ideals in Boolean ring R;

2. If H is an ideal in B, the relations Sa = s and H = B are aeH equivalent are characterized topologically as those Boolean spaces which are con­ tinuous images of an arbitrary representative of R. In particular, the representatives of Boolean rings R' with the following properties:

a. /R' is isomorphic to a subring 3 of a Boolean ring R with

unit e;

b. e is an element of g; are characterized topologically as the totally disconnected continuous

Images of an arbitrary representative of R.

It is a fact in topoligy that a locally compact Hausdorff space

can be imbedded in a compact Hausdorff space. In view of the relations between Boolean spaces and Boolean rings, there are facts of algebraic significance concerning the imbedding of Boolean spaces in compact

Boolean spaces. There theorem which follows reveals the fact that such an imbedding is possible by the adjunction of a single point.

THEOREM VI-20. The non-compact Boolean spaces are characterized

topologically as the non-closed open subsets of compact Boolean spaces;

in particular, every non-compact Boolean space can be converted by the 327 adjunction of a single non-isolated point (in an essentially unique way) into a compact Boolean space. Accordingly, the Boolean rings without unit are characterized algebraically as the non^principal1idealsjin

Boolean rings with unit; in particular, every Boolean ring without unit can be imbedded as a non-principal prime ideal in a Boolean ring with unit (in an essentially unique way).

Tychonoff[197] proved a theorem which suggests a corresponding specialization to the case of Boolean spaces, together with its algebraic interpretation. This theorem in essence says

THEOREM VI-21 (Tychonoff). A topological space is completely regular if and only if it is homeomorphic to a subspace of a compact space.

In the theorems which follow. Stone[191] formulates this special theorem in terms of Boolean spaces of infinite character.

THEOREM VI-22. Let c be an arbitrary infinite cardinal number; let R be an arbitrary class of cardinal number c, let B^ be the class of all characteristic functions C = G(a) defined over R (for each a in

R, either C(a) = 0 or .C(a) = 1; and let R^ be the class of all sets in

B generated from the special sets U , U ,, where U contains all Ç for c ° a a a which C(a) = 0, by the formulation of finite unions and intersections.

By the assignment of each non-empty set belonging to R^ as a neighbor­ hood of every one of its points, B^ becomes a compact Boolean space of character c. The system R^ is a Boolean ring with the set B^ as its unit, with c as its cardinal number and with B^ as one of its repre­ sentative Boolean spaces. 328

THEOREM VI-23. Let be the space defined in the above theorem.

Every Boolean space of character not exceeding c is topologically equivalent to a subspace of B^; and every Boolean ring with unit which has cardinal number not exceeding c is a homomorph of the ring R^.

The last theorem, in particular, is the desired specialization of the theorem of Tychonoff. Some other facts via this theorem are;

THEOREM VI-24. The Boolean spaces B^-{c} obtained from B^ by the suppression of a single point c are topologically equivalent and all are non-compact; the prime ideals in the Boolean ring R^ are isomorphic

Boolean rings without a unit. If any common isomorphism of the prime ideals in R^ is denoted by R*, then every Boolean ring of cardinal number not exceeding c is a homomorph of R*.

The Boolean space B^ is a universal Boolean space due to the fact that every Boolean space of sufficiently small character can be im­ bedded in it. Consequently, the Boolean rings R^ and R* are also universal Boolean rings in the sense that their homomorphisms exhaust all Boolean rings of cardinal number not exceeding c. These rings have the following characterization as universal rings.

THEOREM VI-25. The Boolean rings R^ and R* are isomorphic respectively to the free Boolean rings with and without unit generated by c elements.

The case where c ~ ^ q leads to a very special space known as the

Cantor Discontinuum, which occurs in various areas of mathematics, especially real analysis (see Appendix C).

THEOREM VI-26. The Boolean space B^, c is topologically equivalent to the space D known as the Cantor Discontinuum. 329

The proof of this theorem involves several tools common to analysis and topology. Stone[1911, in the remaining portion of his paper uncovered, many other topological concepts pertinent to Boolean spaces. Though these results are not necessarily algebraic, they,along with the pre­ ceding ones. Inspired others to apply ring theory to analysis, which inevitably led to applications of ideals to analysis.

Applications of Ideals to Continuous Functions

Stone’s papers of 1936 and 1937 opened many "mathematical doors."

For the first time, mathematicians of that decade, 1936-1946, began to take a serious look at continuous functions defined on certain topologi­ cal spaces.

DEFINITION VI-10. Let X be any topological space and let R be the space of real numbers with its usual topology. H*(X, R) is the set of all real-valued continuous bounded functions with domain X and

H(X, R) is the set of all real-valued continuous functions from X to R.

Remark

If K denotes the space of all complex numbers with its usual topology, then H*(X, K) and H(X, K) are defined similarly.

It is very obvious that if X is any topological space, then

H(X, R) and H*(X, R) are commutative rings with unity, where

(f+g)(p) = f(p)+g(p)

fg(p) = f(p)g(p) for all peX; the ring unit is 1, where l(p) = 1 for all peX.

The applications of rings and ideals to continuous functions fall into three general categories; continuous functions on certain 330 topological spaces, entire functions on the space of complex numbers and real-valued continuous functions with Banach space structure. We shall investigate these applications in the very same order, observing the Influence of Stone[191] in each of these categories.

Continuous Functions on Certain Topological Spaces

Silov[188] in 1939 derived some results concerning applications of rings and ideals to the set H(X, K) of all complex continuous func­ tions defined on a metrizable compact space X, The set H(X, K), actually a ring, has its topology defined in the sense of uniform con­ vergence (see Appendix F). In particular, we are concerned with the role of ideals in the ring H(X, K).

Every element f of H(X, K) generates a minimum closed ideal 1(f) containing f. The ideal 1(f) may be defined as follows:

DEFINITION VI-11. The ideal 1(f) is defined to be the inter­ section of all closed ideals I^ containing f or alternately as the closure of the set of all products of the type gf, where g is an arbitrary element of H(X, K).

THEOREM VI-27. The principal ideal 1(f) generated in the ring

H(X, K) by the function f coincides with the set of all functions from

H(X, K) vanishing on the set on which f(x) = 0.

Proof:

Since every function f from 1(f) is the limit function of func­ tions of the type gf, it vanishes at all points of the set N^. We must prove that conversely every function g(x) vanishing at all points of

N^ belongs to 1(f). Consequently, we consider the sequence of all functions g^^x) defined as follows: 331

g^(x) = g(x),

if o(x, N^) > I

g^(x) = n{a(x, N j ) - “ }g(x),

if ^ ^ a(x, N J : I n i n g(x) = 0,

if cr(x, Nj) - ~ • It can be shown that if g(x) vanishes at all points

of the set then the sequence g^(x) converges uniformly to g(x).

Suppose we set g„ (x) ' -F Ô Ô ’ i( o(x, Nj) ; J

h^(x) = 0,

if o(x, Nj) 5 ” .

Consequently, the functions h^(x) are continuous on the whole space X.

Since

g^(x) = h^(x)f(x),

then all functions g^(x) and consequently also the function g(x) belong

to the ideal 1(f). Thus the principal ideal 1(f) generated in the ring

H(X, K) by the function f(x) coincides with the set of all functions

from H(X, K) vanishing on the set on which f(x) = 0.

THEOREM VI-28. All closed ideals of the ring H(X, R) are principal

ideals.

Silov, by way of Theorems VI-27 and VI-28, was able to show that

there exists a one-to-one correspondence between the ideals of the ring

H(X, K) and the closed sets of the space X. To every ideal I corre­ sponds the closed set Nj and I itself is actually the set of all 332 functions in H(X, K) vanishing at every point of the set N^. Further, it is true that the partially ordered set of ideals of the ring H(X, K) with the relation of inclusion as the ordering relation is isomorphic to the partially ordered set of closed subsets of the space X orderedin the opposite manner. Consequently, we come to

THEOREM VI-29. In order that two metrizable compact sets X and

X' be homeomorphic, it is necessary and sufficient that the rings

H(X, K) and H(X’, K) be continuously isomorphic.

In 1939. also, Gelfand and Kolmogoroff[58] published a paper on rings of continuous functions on certain topological spaces. Their paper for the most part dealt with the same topics that were previously investigated by Stone[191] and Silov. The only differences were that they considered a ring of continuous functions defined on a particular topological space as a purely algebraic formation without introducing in it any topological relations And,significantly, the fact that they uncovered some new results. In particular, they considered two rings; the ring H(X, R) of all real continuous functions defined on X and the ring H*(X, R) of all real continuous bounded functions defined on X, where X in both cases is a topological space.

The maximal ideals of the rings H(X, R) and H*(X, R) play a vital role in these new developments of Gelfand and Kolmogoroff.

DEFINITION VI-12. The maximal ideal a is called a point of closure of the set y of maximal ideals if it contains the intersection of all maximal ideals contained in y . 333

Remarks

1. For any ring R, the definition of points of closure makes the set M a topological space.

2. This method of introducing a topology on the set of maximal ideals has already been used by Stone[191].

We shall denote the space y corresponding to the ring H(X, R) by

Y(X) and the space y' corresponding to the ring H*(X, R) by y'(X). If the space X is compact, then the rings H(X, R) and H*(X, R) coincide.

Among the many results proved by Gelfand and Kolmogoroff are

LEMMA 1. If X is compact, then for any ideal a of the ring

H(X, R) not coinciding with the whole ring there exists a point a of the space X at which all functions belonging to a vanish.

THEOREM VI-30. If the space X is compact, then it is homeomorphic to the space y (X).

Proof:

By Lemma 1, it follows that any ideal a not coinciding with the whole ring H(X, R) is contained in a certain ideal 1(a) consisting of all functions vanishing at the point a. Consequently, only ideals of this type can be maximal. It is also a fact that any ideal of the type

1(a) is maximal. Since X is compact it is totally regular and hence the ideals 1(a) and I(a*) corresponding to two distinct points a and a' are different. Therefore, there exists a function in H(X, R) vanishing at the point a and different from zero at the point a'. Thus, correspond­ ing to every point a of the space X, we have the maximal ideal 1(a), obtaining a one-to-one correspondence between the space X and the set y (X) of maximal ideals of the ring H(X, R). 334

We must next prove that the correspondence obtained between X and y (X) is a homeomorphism. Suppose that a set M of points of the space X corresponds to the set y of maximal Ideals. If a point a is a point of closure of the set M, then all points contained in the intersection of all ideals in y , that is, vanishing at all points of the set M, vanish also at the point a. Thus these points belong to the ideal 1(a). By

Definition VI-12 this means that 1(a) is a point of closure of the set

Y. Conversely, if a point a is not a point of closure for M, then by the total regularity of X, we can construct a function in H(X, R) vanishing on M and different from zero at the point a; this function certainly belongs to the intersection of all ideals in Y, but does not belong to 1(a); consequently, in this case 1(a) will not be a point of closure for y« Hence the homeomorphism between the spaces X and Y(X) is thus proved.

THEOREM VI-31. A necessary and sufficient condition for two compact spaces X and X^ to be homeomorphic is that the rings H(X, R) and H(X^, R) be (algebraically) isomorphic.

Through a series of parallel results concerning the rings H(X, R) and H*(X, R), Gelfand and Kolmogoroff conclude that the ring H(X, R) provides us with a more elastic means of investigating the topological properties of the space X than the ring H*(X, R).

In 1948, Hewitt[83] published a paper which like Stone’s paper has become a classic in the area of applications of rings and ideals to analysis and topology. Unlike the paper of Stone[191], this paper explores two of our areas of applications of rings to continuous 335 functions: It Involves continuous functions on certain topological spaces and continuous functions with Banach space structure. We shall exhibit the former area first, saving the latter for the section on continuous functions with Banach space structure.

DEFINITION VI-13. Let X be any topological space. Any ideal I in H*(X, R) or H(X, R) is said to be a free ideal if, for every point peX, there exists an element f of I such that f(p) 4 0.

Remark

An ideal which is not free is said to be fixed.

DEFINITION VI-14. Let X be any topological space and let f be a function in H(X, R). The set of points in X for which f vanishes is said to be the zero set of f and is denoted by Z(f).

Remark

A subset A of X which is the set Z(f) for some f in H(X, R) is said to be a Z-set.

THEOREM VI-32. Let X be a completely regular space (see Appendix

F). Then the ring H*(X, R) contains a free ideal if and only if X is not compact.

Proof t

Suppose X is compact and that a free ideal I exists in H*(X, R).

Then for every psX, there is a function f in I such that f(p) 4 0 and

2 2 consequently f (p)> 0. Since f is continuous, there is an open 2 neightborhood of p, which may be denoted by U(p), such that f (q) >0 for all qcU(p). Since X is compact, there is a finite number of these neighborhoods, say U^, Ug, . . . U^, which will cover the space X. 336

Hence it follows that the function, 2 2 2 2 ^1+4+4+ ' - ' +fn ' 2 where each f^ is associated with the neighborhood U^, is an element of

I which vanishes nowhere and is positive. It is true that such func­ tions on compact Hausdorff spaces have positive lower bounds and possess inverses. This contradicts the fact that I is a free ideal and proves that I must be a fixed ideal.

On the other side of the coin, suppose that X is not compact, and let open covering of X for which no finite subcovering exists. For each peX, we define a function f ,, in H*(X, R) such that p A f , (p) = 1 and f , (q) = 0 for allqeG,' . Such functions f . exist p,A p,A A p,A by virtue of the fact that X is completely regular. The ideal I gen­ erated by the set of functions f ,, where p runs through all the p , A elements of X and A runs through all appropriate indices in A, is cer­ tainly a free ideal. Also it must be a proper ideal. If I were not a proper ideal, then for an appropriate choice of p^, A^, pg, Ag, . . . p^, A^ and functions g^, g^, . . , g^ in H*(X, R), we should have the equality

^ == i=l^Pi, Ai®i where 1 is the function identically equal to unity. This equality implies that the open sets G% , G\ , G% , . . , G% cover X, which is 1 2 3 n a contradiction that to our assumption that there does not exist such a funite subcovering.

Remark

If H(X, R) contains an unbounded function f, then a free ideal in

H*(H, R) may be immediately obtained. For example, the function 337

(f +1) Is positive everywhere, lies in H*(X, R), but clearly bas no inverse in H*(X, R). Thus it generates a proper free ideal.

The fixed maximal ideals in H*(X, R) may be characterized very simply as is displayed in the next theorem proved by Hewitt[83].

THEOREM VI-33. Let X be a completely regular space. The fixed maximal ideals of H*(X, R) are precisely those ideals of the form

E{f|feH*{X, R), f(p) =0}, where p is some fixed point in X; we denote such an ideal by the symbol M^. In the homomorphism defined by M^,

Mp(f) = f(p), for all fEH*(X, R).

Proof :

It is clear that every set M^ is an ideal in H*(X, R). If g 4 OeMp, then g(p) 4 0 and the function g-g(p)eMp. The ideal gen­ erated by Mp and g contains the function Cg-g(p))-g = g(p), and hence this ideal is the whole ring H*(X, R) so that Mp is a maximal ideal.

Conversely, suppose that I is a fixed maximal ideal. Since I is fixed, the set

A = ,gjZ(f) is non-empty and can obviously contain only one point if I is to be maximal. By the same line of reasoning I must contain all of the func­ tions vanishing at the single point in A. Finally in the quotient field , the equalities Mp(f-f(p)) = 0, Mp(f) = Mp(f(p)) = f(p) can be obtained.

A most significant property of rings H*(X, R) is the fact that for every completely regular space there exists a unique compact

Hausdorff space, commonly denoted as BX, with the properties that 338

X C 3X, X = BX and H*(X, R) is algebraically isomorphic to H*(3X, R) as rings. Stone

[191], whose works were discussed earlier in our discourse, was the first to prove the existence and uniqueness of 3X. Also earlier in our discussion of Stone’s works, we pointed out the fact that Stone[191] introduced a topology on the set of maximal ideals in a ring. Since the maximal ideals of H*(X, R) have been clearly identified, a topology on the set of maximal ideals in H*(X, R) can be introduced according to the next theorem.

THEOREM VI-34. Let X be a completely regular space and let M* be the set of all maximal ideals in the ring H*(X, R). Let V^ be the set

E{MeM*, f 4 OeM} and let every V^ be designated as a neighborhood of every maximal ideal M which it contains. As f assumes all possible values, a complete neighborhood system is defined for every point of

M*. The family is closed under the formation of finite intersections. Under the topology so defined, the space M* is a com­ pact Hausdorff space containing a homeomorphic image % of X. The space

X is dense in M* and every function in H*(X, R) and every function in

H*(x, R) can be continuously extended over the whole space M*. The rings H*(X, R) and H*(M*, R) are therefore isomorphic, and M* is the space

3X. The points of M* corresponding to the points of X are the fixed maximal ideals in H*(X,R) and the points of M*nX' are precisely the free maximal ideals of H*(X, R).

As in the case of rings H*(X, R), we can observe that maximal ideals in H(X, R) are indispensable tools in our study. The fixed 339

maximal ideals are easily described by the theorem given immediately

below:

THEOREM VI-35. The fixed maximal ideals in the ring H(X, R) are

the sets of all functions in H(X, R) vanishing at a given point peX, for

a fixed pEX; as before this ideal can be denoted by M^.

The proof of this theorem is the same as the one for fixed ideals

in H*(X, R).

The free maximal ideals in H(X, R), on the other side of the coin,

have many differences, their peculiarities being closely related to

those of the space X, We can observe, in particular, that the family

4'(M), where M is an arbitrary free maximal ideal, can be identified by

purely set theoretic properties. First, we consider the properties of

àÇ(M). (See Appendix F.)

THEOREM VI-36. Let X be any completely regular space. Consider

the following properties of a non-empty subfamily CL of «jf(X) :

1. CL enjoys the finite intersection property.

2. If Aett, BeZ(X), and B3A, then Be«.

3. If WeZ(X) and W 4 OeO, then W ^ a = 0 for some Aett.

4. The intersection of all sets in CL is void.

5. QU contains no sets compact in their relative topologies.

The family & is a family «f(I) for some ideal I in H(X, R) if and

only if 1. and 2. are valid; Û-is the family <^(M) for some maximal

ideal M in H(X, R) if and only if 1., 2., and 3. are obtained; CL is

the familycf(M) for a free maximal ideal in H(X, R) if and only if 1.,

2., 3., and 4. are valid. Finally, if d i s the family d^M) for a free 340 maximal ideal M, then 5. is valid; but the converse is not in general true.

The occurrence of free maximal ideals may be described as follows:

THEOREM VI-37. Let X be a completely regular space. An element f of H(X, R) lies in some free maximal ideal if and only if Z(f) is not compact.

Proof:

If f is an element of a free maximal ideal, then Z(f) is not com­ pact by the last theorem. On the other hand, suppose Z(f) is a non­ compact subset of X and let be an open covering of Z(f) admitting no finite subcovering. For every point peZ(f) and every A such that peG, , let , be a function in H(X, R) such that . (q) = 0 for all q in G. ,. p,A A Since X is completely regular, such functions exist. The set of

functions fUfO where p runs through all points of Z(f) and A p , A

through all indices such that peGA, generates a free ideal é. c5 must be proper also because if ^ is improper then the relation

' ■ A, must be valid for appropriate (A^, p^), . . . (A^^ p^) and functions

g]^, ^2 * . . , g% and g in H(X, R). There is, however, a point in Z(f) which lies in (G, M G, U G, u . . . U G, )', and at this point the Ai Ag A3 '-' Aj^

function

must vanish, contradicting the above relation. Hence é is not the 341 whole ring H(X, R); then by Zorn's lemma, it can be proved that tP is contained in a (free) maximal ideal, which must contain f.

Hewitt[83] points out the various topologies in H(X, R) and

H*(X, R) in the definitions which follow. We require these definitions in some of the results which will follow the definitions.

DEFINITION VI-15. Let X be any topological space and let M be any metric space. Let the set of all continuous mappings of X into M be denoted by H(X, M). The set H(X, M) may be made into a topological space, where neighborhoods U^(f) are defined as E{g|geH(X, M), |g(p)- f(p)| < n for all peX , n being an arbitrary positive real number and f being an arbitrary element of H(X, M)}-. The resulting topology is called the u-topology. H(X, M) may also be made into a topological space by considering an arbitrary compact subset K of X and any positive real number n. The neighborhood U„ • (f) is then defined as E{glgeH(X, M), iv J n |g(p)“f(p)i < n for all peK}. The topology in H(X, M) which results by taking arbitrary sets K and arbitrary positive real numbers n is called the k-topology. Finally, H(X, M) may be made into a topological space by substituting arbitrary finite subsets of X for arbitrary compact subsets of X in the definition of the k-topology. The resulting topology is called the p-topology. Bounded functions in H(X, M) being defined in the usual way, and the set of all such functions being denoted by

H*(X, M), the U - , k-, and p-topologies in H*(X, M) are defined as the relative topologies for H*(X, M) as a subspace of H(X, M).

DEFINITION VI-16. If X is an arbitrary topological space, the set H(X, R) may be made into a topological space by considering an arbitrary function IIeH(X, R) having the property that II(p) is positive 342 for all peX, For any such function ir and function f in H(X, R) , the neighborhood U^(f) is defined as E{g|geH(X, R) [g(p)-f (p) | < tt(p ) for all peX}. The resulting topology is called the n-topology.

Remarks

1. The u-topology appears to be the most natural in considera­ tions involving rings H*(X, R).

2. The TT-topology appears most natural in the cases involving the rings H(X, R).

3. The k” and p-topologies are not used very often in connection with the rings H(X, R) and H*(X, R).

4. In topological spaces on which every continuous real-valued function is bounded, the u-topology and the -topology coincide.

5. In the case of compact topological spaces X, the k-, u- and

IT-topologies coincide.

We shall utilize the u-topology in a discussion of some further properties of ideals in the rings H(X, R). Hewitt[83] proved

THEOREM VI-38. Let X be an arbitrary topological space. Under the u-topology with

(f+g)Cp) = f(p)+gCp),

fg(p) = f(p)g(p)j

(af)(p) = a(f(p)), for all peX and o e R, the set H(X, R) is a commutative ring over R with unit 1, where l(p) = 1 for all peX, in which the operations of addition and multiplication are continuous, the set of elements with inverses is an open set and the inverse is a continuous operation wherever defined.

If there exists an unbounded function in H(X, R), then H(X, R) is not 343

metrizable (see Appendix F) under the u-topology and. Indeed, falls at

every point to satisfy the first axiom of countability. If every func­

tion in H(X, R) is bounded, then the u-topology and the u-topology

coincide in H(X, R).

THEOREM VI-39. Let X be any completely regular space. The 'n'-

closure of any ideal I in H(X, R) is again an ideal in H(X, R).

Proof:

Let f and g be any elements in I (throughout our proof, the

closure operator is in terms of the u-topology) and let U^(f+g) be an

arbitary u-neighborhood of f+g. In view of the last theorem, there

exist neighborhoods U^(f) and Up(g) such that feU^(f) and geU^(g) imply

that f+geU^(f+g). Since f and g are in I, then U^(f+g) contains an

element fQ+g^, Where f^ and g^, and hence f^+g^ are in I. This implies

that f+gcl. Now if fel and is any function in H(X, R) and U^(^f) is

an arbitrary neighborhood of i(>f, then as above there exists a neighbor­

hood U^(f) such that feU^(f) implies that iJ/feU^(iljf). This implies that

i|jfel and therefore I is an ideal in H(X, R).

THEOREM VI-40. Every maximal ideal in H(X, R) is u-closed.

Proof :

If M is a maximal ideal in H(X, R), the closure M must be either

M itself or the ring H(X, R) as was indicated in the last theorem

(Theorem VI-39). If the latter is true; that is,

M = H(X, R),

then there would be elements of M in every neighborhood of 1 ; since 1 has an inverse, and the set of elements with inverse is open in H(X, R), 344

this implies that M contains an element with an inverse. This is

impossible if M is to be a proper ideal, and

M = M

is the only remaining possibility. Therefore, every maximal ideal in

H(X, R) is IT-closed.

THEOREM VI-41. Let A be any non-empty subset of the completely

regular space X, Then the set of all functions vanishing on A is an

Ti-closed ideal in H(X, R).

Previously we have discussed separately the ideals in H(X, R) and

H*(X, R). In particular, we have discussed the maximal ideals in these

rings. However, we have not really made a comparison. The next

theorem does just that via the set aC(M).

THEOREM VI-42, Let X be a completely regular space and let M* be

any maximal ideal in H*(X, R). Let 6 (M*) be the family of all sets

N(|f|-e), where f runs through all the elements of M* and e runs through

feM* and Z(f) is non-empty. Then there exists a unique maximal ideal M

in H(X, R) such that .^C(M) = 6 (M*); for all feH*(X, R), f is an element of M* if and only if M(f) is 0 or is infinitely small in •

Conversely, if M is any maximal indeal in H(X, R), the set U of elements

feH*(X, R) such that N(|f|- e ) e 5((M) for all positive real numbers e con­

stitutes a maximal ideal in H*(X, R). If n is an arbitrary maximal

ideal in H(X, R), then the set U of elements feH*(X, R) such that

N(IfI- e ) e^(M) for all positive real numbers e constitutes a maximal

ideal in H*(X, R). If n Is an arbitrary maximal ideal in H(X, R) , then

nHH*(X, R) is contained in precisely one maximal ideal of H*(X, R) 345

and nnH*(X, R) is a maximal ideal in H*(X, R) if and only if n is a

real ideal. (See Appendix A.)

Hewitt's results generated a great deal of interest in rings of

continuous functions and their ideals. This is particularly true for

the maximal ideals. In 1954, Gillman, Henriksen,and Jerison[62] pub­

lished a paper on a theorem which was proposed by Gelfand and

Kolmogoroff and applied their results to some problems previously con­

sidered by Hewitt.

THEOREM VI-43 (Gelfand-Kolmogoroff). A subset M of H(X, R) is a

maximal ideal of H(X, R) if and only if there is a unique point pegX

such that M coincides with the set

M^ = {f|f£H(X, R), peZ(f)}.

Proof:

In order to prove that M^ is a maximal ideal of H(X, R), it

suffices, according to Hewitt's results (Theorem VI-42 above), to show

that the family ^(M^) is a maximal subfamily of<^(H(X, R)) having the

finite intersection property and not containing the empty set. The

last condition is obviously satisfied. It has been proved in the paper

by Hewitt[83] that any two disjoint zero-sets are completely separated

and by Cech[20] that any two completely separated subsets of X have

disjoint closures in 3X. This implies that ^(M^) has the finite inter­

section property. In order to establish maximality with respect to

this property, consider any geC for which Z(g) meets every member of

.^(M^), we must prove that geM^. Now let S2 be an arbitrary neighborhood

of p in 3X and let % be any neighborhood of p in 3X such that Zcîï. 346

But by the normality of PX, there is an feH*(X, R) such that

f(ï) = 0, f(3X-fi) = 1.

Since X is dense in 0X, the set 2 0x is not empty and its closure

contains p, and we have f (Z AX) = 0. Hence peZ(f), so feM**. Therefore

Z(g) intersects Z(f). Since Z(f)cfl, and ÎÎ was an arbitrary

neighborhood of p, then this implies that Z(g) meets every neighborhood

of p. Consequently peZ(g); that is, gcM**,

Conversely, let M be any maximal ideal of H(X, R). The family i(M) has the finite intersection property and does not contain the

empty set since M is a proper ideal. Therefore, since BX is compact,

the intersection

A(M) = A z(f) fem

is not empty. Clearly for each peA(M), McM**. Since M** is an ideal,

as has been shown, and M is a maximal ideal, we have that

M = M?.

Since the space X is completely regular, then the uniqueness of p can be deduced.

Prime ideals play a very important role in the theory of ideals.

Thus far we have not cited any results concerning these ideals in the

rings H*(X, R) and H(X, R). We shall pause here to investigate the

interplay between prime ideals and conditions on the topological space

X according to Gillman and Henriksen[63],

DEFINITION VI-17. For every point p of BX, we denote by N** the

ideal of H(X, R) consisting of all feH(X, R) such that Z(f) contains

an X-neighborhood of p. 347

THEOREM VI-44. If P is any prime ideal contained in the maximal ideal M**, then PoN**.

Proof;

Let feN**. By definition of N**, there is an open subset of gX such that peU = n H x and f(U) = 0. Since gX is completely regular, there is a geH*(X, R) such that g(p) = 1, g ( g X - n ) = 0. Clearly fg = 0.

Since P is prime and g^M**, we must have feP. It follows that N**cp,

THEOREM VI-45. For any completely regular space X, every prime ideal P of H(X, R)(P* of H*(X, R)) is contained in a unique maximal ideal of H(X, R)(H*(X, R)).

Corollary. The residue class ring of H(X, R) with respect to any prime ideal P contains exactly one maximal ideal namely

p ^ ; this ideal consists of all non-units of — . The appropriate corresponding statement holds for H*(X, R).

THEOREM VI-46. For every point p of gX, the following two statements are equivalent:

1. M** is the only prime ideal containing N**.

2. M** = N**; that is, every continuous function f such that peZ(f) vanishes on an X-neighborhood of p.

Gillman and Henriksen[63] made many successful investigations into the ideal structure of rings of continuous functions. In 1956, they uncovered the conditions necessary on rings of continuous functions in order for every finitely generated ideal to be principal. 348

DEFINITION VI-18, A commutative ring S with identity is called an F-ring if every finitely generated ideal of S is a principal ideal,

DEFINITION VI-19. A completely regular space X such that

H(X, R) is an F-ring is called an F-space.

Remarks

1. Every discrete space is an F-space.

2. Every homomorphic image of an F-ring is an F-ring.

THEOREM VI-47. Let Y be a subspace of an F-space X such that every element of H(Y, R) has a continuous extension to X. Then Y is also an F-space.

DEFINITION VI-20. Two subsets A and B of a space X are said to be completely separated if there is a function keH(X, R) such that k(A) = 0 and k(B) = 1; also A, B are completely separated in this case.

Remarks

1. Cech[20] showed that completely separated subsets of X have disjoint closures in 8X.

2. Urysohn's Theorem states that any two disjoint closed subsets of a normal space are completely separated. (See Appendix F.)

The next theorem characterizes rings of continuous functions in which every finitely generated ideal is principal. This theorem involves the function ]fj which has the following algebraic importance: |f| is 2 2 2 the unique element g such that g = f , and g+u is a unit for every unit u.

THEOREM VI-48. For every completely regular space X, the following statements are equivalent: 349

1. X is an F-space; that is, every finitely generated ideal of the ring H(X, R) is a principal ideal.

1*. BX is an F-space; that is, every finitely generated ideal of the ring H(gX, R) or H*(X, R) is a principal ideal.

2. For all f, gcH(X, R), the ideal (f, g) is the principal ideal (ifi+Ul).

3. For all feH(X, R), the sets P(f), N(f) or P(f), N(f) are completely separated.

4. For all fsH(X, R), f is a multiple of |f|; that is, f = k|f| for some kcH(X, R).

5. For all feH(X, R), the ideal (f, |f|) is principal.

The following results concerning the Stone-Cech compactification are also valid as auxiliary results. (See Appendix F.)

2*. For all t|), il)eH(BX, R) or H*(X, R), the ideal (4i» $) is the principal ideal

3*. For all eH(BX, R), the sets P() are completely separated.

4*. For all c()eH(BX, R), * is a multiple of

Corollary. Any point of an F-space at which the first axiom of countability holds is an Isolated point.

Remark

A metric space is an F-space if and only if it is discrete.

THEOREM VI-49. A completely regular space is an F-space if and only if for every maximal ideal M of H(X, R), the intersection of all 350 prime ideals contained in M is a prime ideal; that is, if and only if for every point peBX, the ideal N** of H(X, R) is a prime ideal.

THEOREM VI-50. A completely regular space X is an F-space if and only if for every zero-set Z of X every function geH*(X-Z, R) has a continuous extension heH*(X, R).

THEOREM VI-51. For every locally compact, a-compact space X,

BX-X is a compact F-space.

Remarks

1. The space of reals R is not an F-space.

2. BX*-R* is a compact F-space which is also connected.

Kohls[102] added another dimension to the theory of ideals in rings of continuous functions. In 1958, he published a paper which in addition to the rings we have studied previously, such as H(X, R) and

H*(X, R) for various topological spaces X, he considered Hg(S, R), the subring of all functions in H(X, R) with compact supports and H^(X, R), the subring of all functions of H(X, R) which vanish at infinity.

LEMMA VI-2. Let X be a completely regular Hausdorff space. Let

A be any closed subset of BX. Then the set I consisting of all feH(X, R) for which Z(f)^ contains a neighborhood of A is the smallest ideal of H(X, R) such that A(I) = A.

LEMMA VI-3. Let X be a completely regular Hausdorff space. Then the ring H^(X, R) is the intersection of the free maximal ideals

(see Appendix A) of H*(X, R).

In view of Lemmas 2 and 3, Kohls arrived at the result: 351

THEOREM VI-52. Let X be a locally compact Hausdorff space and let a be an ideal of H*(X, R). Then the following statements are equivalent:

1. H(X, R) e a S H^(X, R).

2. For all M^(H*(X, R)), M -2 a if and only if M is a free ideal.

3. The mapping

p -> M* n a* PeX is a homeomorphism from X to M^Ca)» where M^(H*(X, R)) is the Stone[191]

topology discussed at the beginning of this chapter and Mg(a) is the

Stone topology on H*(X, R) and a respectively.

Entire Functions on the Field of Complex Numbers

It is well known in analysis that differentiability implies continuity. In view of this fact, we should like to investigate the

ideal structure of certain rings of differentiable functions. In view of our first statement we should expect some similar results. Entire

functions fall into the category of differentiable functions on the

complex plane.

Helmer was among the first mathematicians who extensively

investigated the algebraic structure of the ring of entire functions.

In his paper of 1940 Helmer[78] explored the divisibility properties of

integral functions. (The term integral function was a term used

primarily by the British for transcendental entire function. See

Appendix C.)

The fields of rational, real or complex numbers will be denoted by Q, R or C, respectively, as usual. Also in keeping with the standard notation, K[z] represents the domain of polynomials in z with 352 coefficients in K, where K is some arbitrary subfield of C, Z^, again, denotes the set of all zeros of f(z) each occurring a number of times equal to its multiplicity. Finally, K will be used to denote the domain of integral functions in z with coefficients in K, such that the integral functions are all power series

f(z) = Za z^, 1 with a in K and lim a n = 0 and K* will denote the subset of those n n-Ko n f(z) in K with ord f < « (see Appendix C).

Some significant results pertinent to K and K*, which were proved in the paper of Helmer[78], follow;

THEOREM VI-53.. An element of K is a unit of K if and only if it has the form ce^^^^, with c ^ 0, in K, f(z) in K and f(0) = 0.

THEOREM VI-54. An element of K* is a unit of K* if and only if it has the form ce^^^^ with c # 0, in K, and F(z) in K[z] and

F(0) = 0.

THEOREM VI-55. Let K be an algebraic number field (see Appendix

E); then for every zero-set Z, there is a function f(z) in K such that Z = Zg. Moreover, if Z is symmetric, such a function can already be found in Q(z) (see Appendix C).

THEOREM VI-55 can be used to deduce a result which is in agree­ ment with the integral functions which we encountered in Chapter III in connection with Kronecker's notions of ideal theory.

THEOREM VI-56. Any finite or Infinite set of functions of K has a greatest common divisor in K. 353

Proof:

Let Z be the intersection of the zero-sets of the functions

f(z) under consideration. If K is real, every Z^, and hence Z, is

symmetric. Therefore, by the above theorem (Theorem VI-57)» no matter whether K is real or imaginary, there will be a function d(z) in K

for which Zj = Z. This function is obviously the greatest common divisor of the functions f(z). It is uniquely determined except for an

arbitrary unit factor in K.

THEOREM VI-57. Q, and hence every K, contains ideals which

are not principal ideals.

THEOREM VI-58. Every ideal in K that possesses a finite basis

is a principal ideal.

In general, the theorems of the classical ideal theory in fields

of algebraic numbers hold in rings of analytic functions on compact

Riemann surfaces. Schilling[184], in his investigation of ideal

theory on open Riemann surfaces, proved several important results con­

cerning rings of entire functions. He used Z to represent the complex

number sphere whose points at finite distance p = p^ are in 1-1 corres­

pondence with the elements a of the complex number field C. With each

point p, he associated a model Z^ of Z and considered the functions

f(z) on Z with values in {Z^}; that is, f(p)eZp for peZ. As usual we

say that a function f(z) is meromorphic on Z if there exists for each

point peZ a convergent expansion

' iïmS, i‘p •

where the first nonvanishing coefficient C has a finite integral p , m 354 subscript. If we omit from S the point p^, we obtain the open complex number plane 2 ' whose points are in 1-1 correspondence with the numbers of C.

Schilling used F(z) to represent the field of all rational func­ tions of a complex variable z. He used also F(Z') to represent the field of all meromorphic function (see Appendix C). It has been shown that the field F(Z') contains as a subring 0(20» the integral domain of all entire functions f(z) with Vp(f(z)) iO for all peZ', where Vp(f(z)) is the valuation of f (see Appendix A).

DEFINITION VI-21. An ideal a of 0(2') is an additive subgroup of

0(2') which admits the elements of 0(2') for multipliers.

Helmer[78] has already pointed out the fact that the ideal theory of C[z] has no direct analogue in 0(2'). The ideal theory of C[z], according to Schilling[184], has two characteristic properties: there are no properly ascending chains of ideals and secondly there are no other prime ideals but the ones corresponding to points of the algebraic curve or Riemann surface. It can be shown that the ring

0(2') represents an entirely different arithmetic structure. Consider the entire function f(z) = sin nz with the zeros ±n and set fj^(z) = sin iTZ, fg(z) - , £3 (2 ) = . . . , ° z(z -1 )

sin uz . Then f has the zeros ±(n+l) . . . and z(z^-l^) . . . (z^-n^) f^(z) = *z(z^-l^) . . . (z^-n^). Then observe that 0(Z')f^<

0(2'' • • is a properly ascending infinite chain of divisors. 355

Due to the properties of O(E') as a ring it becomes necessary to modify the definition of an ideal in order to develop an ideal theory on O(Z') such that resulting theory resembles the ideal theory of fields of algebraic functions or algebraic numbers. There are two methods of approach which were utilized by Schilling: first a recast­ ing of the definition of ideal and second, the technique of r-ideals and quasi equality. In order to carry the first approach, we require a topology on O(Z').

DEFINITION VI-22. Suppose that {f^^ is an infinite sequence of entire functions. We define f ^ f as n ->■<*> to mean uniform convergence in any bounded region of Z.

DEFINITION VI-23, An ideal a of F(Z’) with respect to O(Z') is called closed if:

1. a is closed under addition and under multiplication by elements of D(Z'); that is, o is an ideal in the algebraic sense;

2. There exists a nonzero integral function f such that f(a) ^

O(Z');

3. a is closed in the topology of the first definition (VI-22).

Remarks

1. Part 3. of the definition implies that a contains all conver­ gent infinite sums

for which a^ea.

2. Let va = min V a = V a, all peZ'. The vector va has at most aea p p a denumerable infinitude of components V^a distinct from 0 since an element aea has at most a countable set of zeros in Z*. 356

LEMMA 4. If a is closed and va = {0}, then a = O(Z').

LEMMA 5. Let a, B be two closed ideals and define the product a*B as the totality of all convergent sums

Z a b , a ea, b eB. . n n n ' n n=l Then a*B is closed.

LEMMA 6. All closed ideals a are principal.

In view of these lemmas, we have

THEOREM VI-59. The closed ideals a of F(Z') with respect to

O(Z') form a multiplicative group which is isomorphic to the vector group V.

Turning to the second approach, which involves quasi-equality, we have

DEFINITION VI-24. Two ideals a^, a^ of F(Z') with respect to

O(Z') are called quasi-equal, written ~ (^2 ' ^“i “ ^“2 ’

Remark

Quasi-equality is an equivalence relation.

LEMMA 7. Let a be the closure of the ideal a, in the sense of definition VI-23; then va = va.

It can be shown that the functions f of F(Z) of which are holo- morphic at p form a ring R^, the so-called valuation ring of p in F(Z).

This particular ring contains a single prime ideal M* which consists of all functions which vanish at p. This prime ideal is obviously closed since it consists of entire functions which vanish at p. We consider some of the other properties of O(Z').

LEMMA 8, All prime ideals of O(Z') are maximal. 357

LEMMA 9. A maximal ideal M of O(Z') is nonclosed if and only if

O(E') — —— is a proper extension of the complex number field C.

LEMMA 10. There exists no valuation V of rank one such that a nonrclosed prime ideal M of O(S') is equal to Pno(Z'), where P is the prime ideal of the valuation ring of V in F(S').

THEOREM VI-60. The points of S' are in one-to-one correspondence with the closed prime ideals of the ring O(S') of analytic functions on

S'.

LEMMA 11. Each closed ideal a of 0*(T) without zeros on T is equal to 0*(T), where 0*(T) is the ring defined below.

DEFINITION VI-25. 0*(T) is a ring of real (or complex) valued functions continuous on a space T with a compact open topology, defined by the agreement that f^-^ f in 0*(T) if the sequence f^ approaches f uniformly on the compact subset of T.

THEOREM VI 61, The closed maximal ideals M of 0*(T) are in one- to-one correspondence with the points of T.

Proof;

Suppose p is a point of T; then all feO*(T) with f(p) = 0 form a closed maximal ideal M^. Conversely, the functions f of a maximal closed ideal M must have a common zero, p, for otherwise M = 0*(T) by

Lemma 11. Let be the set of all functions vanishing at p. Then

M 5 a and consequently M = a . The functions a cannot vanish at P ^ P P another point q since the complete regularity implies the existence of feOp with f(p) = 0, f(q) f 0.

The maximal ideals M* of the ring 0*(S') of all continuous complex valued functions on S' can be related to the maximal ideals M 358

of O(S'). For each maximal ideal M let the set M*cO*(2’) consist of

all functions feO*(S’) for which there exists an element mcM such that

f vanishes at all zeros of m. The set M* is an ideal. By definition

of an ideal fgeM* for any geO*(S'), Now if f^ is another function of

M* with a corresponding function then f+f^ vanishes at all zeros

of a function h in M whose zeros are those common to m and m^^. Note

that h may be constructed by the method of Lemma 8, Furthermore, M*

is a maximal ideal. Suppose (M*, g)c 0*(S'). We pick any m f 0 in M.

Then m and g must have zeros in common. For otherwise k = mm+gg is a

function of (M*, g) without zeros, so that k ^e0*(2') and 1 = k

g). This is a contradiction. Let n be an entire function whose zeros

are the common zeros of m and g. Then g vanishes at all zeros of n and neM. Hence geM*.

Conversely each M* determines a prime ideal M*no(Z') of O(E');

this ideal is maximal by Lemma 8. Using the original correspondence

we obtain M*riO(T') = M,

Thus far in our discussion of ideal theory and its application to entire functions, we have restricted our discussion to the results of

Helmer pertinent to transcendental entire functions and Schilling's

theory of the ring of entire functions on open Riemann surfaces. How­ ever, at this point, we should like to explore some of the more general properties of entire functions in the same vein as that established by

Henrlksen[79].

The ideals in the ring H(C, C) of entire functions are classified in the same manner as those in the ring H(X, R). In particular, the 359

Ideals 1^ are called fixed if every function in them vanishes at at least one common point; otherwise the are called free Ideals. In general most of the results which were true in the case of the ring H(X, R) of the section entitled "Application of Ideals to Continuous Functions on

Certain Topological Spaces," are true for the ring H(C, C). However, there are some very significant facts concerning maximal ideals, their residue class fields and prime ideals.

DEFINITION VI-26. If feH(C, C), then a(f) = {zeC|f(z) = 0}.

DEFINITION VI-27. If I is any subset of H(C, C), let a(I) 5 {a(f)|fel}.

Remarks

1. If f is any non-zero element of H(C, C), then a(f) is a closed, discrete subset in the natural topology of C.

2. If D is any discrete, closed subset of C, then there is an feH(C, C) such that D = a(f).

THEOREM VI— 62. Every maximal fixed ideal of H(C, C) is of the form

■ I(ZQ) = {feH(C, C)|f(zg) = 0} for some z^eC. Moreover, the residue class field of every maximal fixed ideal is the complex field C.

Proof;

If Zq is any fixed element of C, then we can let I(Zq)={fEH(C, C)| f(z^) = 0}. Clearly I(z q ) is a fixed ideal of H(C, C). Moreover, the mapping f(z) *> f(zg) is clearly a homomorphism of H(C, C) onto C whose kernel islCZg). Hence I(Zg) is a maximal fixed ideal. On the otherhand, if I is a fixed ideal and if g^a(f) contains two points 360 z^, Zg» not necessarily distinct, then I is properly contained in I(z^) or iCzg). Thus the theorem is valid.

The maximal free ideals of H(C, C) are not as simple in structure as the maximal fixed ideals. The next theorem exemplifies this.

THEOREM VI-63. A free ideal M of H(G, C) is maximal if and only if a(M) satisfies the following; If D = ^^n^n=l ^ny infinite, closed, discrete set of C such that BCiaCf) is nonempty for every feM, then Dea(M).

Proof; = >

Suppose M is a free ideal and the above statement holds. If M is not maximal)then there is an ideal N properly containing M. Suppose geN and Aag(N) is empty. Then a(g)Ha(f) is nonempty for every feN fea(N) ^ and hence for every feM. Hence by the above statement, geM. Hence M is a maximal free ideal.

Proof; < =

Suppose M is a maximal free ideal. If there were an infinite, closed, discrete set D violating the above statement, then any gcH(C, C) such that (x(g) = D would generate together with M an ideal N properly containing M. Hence the above statement must hold.

Since maximal free ideals are rather complicated in structure, then it is natural to expect that their residue class fields are rather complicated. p THEOREM VI-64. If M is a maximal free ideal, then — contains a M subfield isomorphic to the field H(z) of all rational functions of a complex variable z. 361

Corollary. The field C is a subfield of . If H(C, C) is H (C C) considered as an algebra over C, then the residue class field ---^--- may be considered as a division algebra containing C as a proper subfield. HfC O') THEOREM VI-65. If M is a maximal free ideal, then ---^--- is isomorphic to C as a ring.

DEFINITION VI-28. a*(f) is used to represent the sequence of distinct zeros of f arranged in some order of increasing modulus.

It is true that if I is an ideal of H(C, C), then a(I) has the finite intersection property.

DEFINITION VI-29.

1. If a*(f) = {a^}, then we can represent the multiplicity of a^ as a zero of f by 0^(f).

2. If a is a nonempty subset of a*(f), we use 0^(f:a) to represent the function 0^(f) with domain restricted to a.

3. Let m(f) = sup 0 (f), if f 0. Let ra(0) = a. n>l “ Now that our basic definitions are established, we can character­ ize the nonmaximal prime ideals of H(C, C) according to Henriksen[80].

THEOREM VI-66.

1. There exist nonmaximal prime ideals of H(C, C).

2. A necessary and sufficient condition that a prime ideal P of H(C, C) be nonmaximal is that m{f) = “ , for all feP.

Proof; (For 1)

Let

S = {feH(C, C)|m(f) < »}. 362

Clearly S Is closed under multiplication and does not contain 0. If g 0 is in H(C, C)-S, g is contained in a prime ideal P not intersecting S by McCoy[153]. Since any maximal ideal contains an f such that m(f) = 1, according to Henriksen[79], P cannot be maximal.

Proof: (For 2)

The sufficiency is clear from the above. If feP with m(f)< “ , the primeness of P ensures that there is a geP with m(g) = 1. Suppose the maximal ideal M contains P and let heM. But by Helmer[78], there is a deM such that

a(d) = a(g) ria(h) .

Now g = g^d, where «(g^)^«(d) is empty, since M(g) = 1. Since P is prime, it follows that either g^^eP or deP. But M 4 H(C, C), so that g^ is not in P. This implies that d and hence h is in P; therefore,

P = M.

Corollary. Any prime, fixed ideal of R is maximal.

THEOREM VI-67. Every prime ideal P of H(C, C) is contained in a unique maximal free ideal M.

THEOREM VI-68. If M is a maximal free ideal of H(C, C), then

p* = is a prime ideal and is the largest non-maximal prime ideal contained in M.

We should like to look at the residue class rings of prime ideals. We make use of the definition of a valuation . ring given by

Krull[108]. 363

DEFINITION VI-30. An integral domain D such that if f, geD, then f divides g or g divides f is called a valuation ring.

Remark

A valuation ring possesses a unique maximal ideal consisting of all its nonunits.

LEMMA 12. If peP^, the set of all prime ideals contained in M, then f is singular modulo P (f 5 l(mod P) if and only if feM.

THEOREM VI-69. The residue class ring - of a prime ideal

P of H(C, C) is a valuation ring whose unique maximal ideal is principal.

Proof;

By the lemma above, Lemma 12, every element of H(C, C)-M is a unit, so we can assume that f, geM. Let

a(d) = a(f) na(g) , so that ct(-^) ria(-^) is empty. At least one of ^ H(C, C)-M, and H(C C) hence is a unit modulo P. Therefore — — is a valuation ring.

2 f If, in particular, f is chosen to be in M-M , ^ cannot be in M, so that g is a multiple, modulo P, of f. Therefore the unique maximal M HfC C) ideal ^ of p— - is generated by f , and hence is principal.

THEOREM VI-70. The residue class ring of a nonmaximal prime ideal P is Noetherian if and only if P = P*, where P* is the intersection of all powers of a maximal free ideal M.

DEFINITION VI-31.

1. If the nonunits of a Noetherian ring D with unity form a maximal ideal M such that 364

“ k n M = (0), k=l D is called a local ring.

2. If fg, . . is a minimal basis for M such that f^,

fg, . . , generate a prime ideal (i = 1, 2, . . , n), D is called

a regular local ring.

3. Using the powers of M as a system of neighborhoods of 0, we

call D complete if. every Cauchy sequence in D has a unique limit.

LEMMA 13. The residue class ring ^ is complete.

The above lemma is utilized to prove p THEOREM VI-71. The residue class ring ^ is isomorphic to the

ring C{z) of all formal power series over C.

At the beginning of section entitled "Applications of Ideals to

Continuous Functions," when we defined the ring H(X, R), we could have

easily defined this structure to be a Banach algebra because of its basic properties. Since a Banach algebra is a ring, we shall look at

the ideals associated with H(X, R) in terms of its Banach algebra

properties as well as some ideals of other Banach algebras.

Continuous Functions with Banach Space Structure

In Chapter II, in connection with Hedderbum's results, we

encountered the concept of an algebra. At that time, we pointed out

that an algebra is a special kind of ring. There are several very

important algebras with additional properties that claim analysis as

their home base. In particular, a Banach algebra is a special algebra which is peculiar to both complex and real analysis. At this point,

we make some formal definitions. 365

DEFINITION VI-32. A Banach algebra A is an algebra such that the underlying linear space of A is a Banach space with the norm of x e A denoted by ||x|| such that the inequality

l|xy|| - l|x|| ||y|l holds for all x, y in A.

The ring H*(X, R), which was encountered earlier, is a Banach algebra over R. The norm in this space is defined by

||f II = sup |f(p) I. peX

The linear space properties are derived from the fact that

(f+g)(p) = f(p)+g(p)

(fg)(P) = f(p)g(p)

(af)(p) = a(f(p)) for all peX and aeR. It can be shown that fg = gf and jjljj = 1.

Suppose for every e >0, there is a peX such that jjfgjj - E<|f (p)g(p) |.

But |f(p)g(p)| = |f(p) I ' |g(p) I - [jfjj ||g|| Î that is^ jjfgjj - E £ jjf |j ijgjj for every e>0; this implies that

jjfgil ^Ijfli jjgll .

Upon studying the rings of functions H*(X, R), it is inevitable that we come to grips with the question: Suppose A is a Banach algebra over R; under what conditions is H*(X, R) =A for some space

X? The next theorem addresses this very question.

THEOREM VI-72. Let A be any Banach algebra over a field R.

There exists a compact Hausdorff space X such that A can be mapped onto

H*(X, R) by a norm-preserving algebraic isomorphism if and only if the following conditions are valid in A: 366

1. (x^+e) ^ exists for all xeA, e being a unit in the neighbor­ hood system E.

2. (e [|x|[ ^-x^) ^ exists for no xeA.

The ideal structure of H*(X, R) as a ring has been given in the section entitled "Continuous Functions on Certain Topological Spaces," earlier in this chapter. We should like, however, to explore the ideal structure of the Banach algebra H(D, C).

H(D, C) is the collection of all complex functions which are continuous on the closed unit disk D, where

D = {zeCI]zl 5 1} and analytic at each interior point. Hoffman[88] points out the fact

H(D, C) is a Banach space under the sup norm

lUIL = sup |f(z)| |z|5l and the fact that each feH(D, C) is the Poisson integral of its boundary values:

£(re^®) =

(see Appendix C).

The ideal theory of any ring is known in general if we know the maximal and primary ideals and facts about their intersections. In particular, if R is a commutative ring with unity, we know the ideal theory of R if we know,1) what the maximal ideals of R are, 2) if every proper ideal is an intersection of maximal ideals, 3) what the primary ideals of R are, 4) if every ideal is an intersection of primary ideals. In the case of a Banach algebra, however, we need to know what the closed ideals are. 367

Suppose z^{zj|z| - 1}. Then {feH(D, C)|f(z) = 0} is a maximal ideal in H(D, C). Suppose z^H(D, C) = {feH(D, C)jf(O) = f’(0) = 0}. 2 Then it can be shown that z H(D, C) is a closed ideal of H(D, C). We should like to see what other ideals are closed and what they are like.

First, however, we require

DEFINITION VI-33. An inner function is an analytic function g in the unit disc such that

|g(z)| t 1 and

|g(e^®)| = 1 almost everywhere on the unit circle.

Hoffman[88] supplies the following results;

THEOREM VI-73. Let K be a closed set of Lebesgue measure zero on the unit circle. Let F be an inner function such that

1. If a^, ag, ag, . . . are the zeros of F in the open disc, then every accumulation point of the a^ is in K.

2. The measure determining the singular part of F is supported on K. Let J be the set of functions of the form Fg, where g is a func­ tion in H(D, C) which vanishes on K. Then J is a closed nonzero ideal in H(D, C).

On the other hand, we have

THEOREM VI-74. Let J be a nonzero closed ideal in H(D, C) and let K be the intersection of the zeros of the functions in J on the unit circle. Let F be the greatest common divisor of the inner parts of the nonzero functions in J. Then J is precisely the set of func­ tions of the form Fg, where g ranges over the functions in H(D, C) which vanish on K. 368

Corollary 1. Every maximal ideal of H(D, C) is of the form

= {feH(D, C)|f(X) = 0} for some point X in the closed unit disc.

Corollary 2. If f^, fg, fg, . . ^ f^^ are functions in H(D, C) which have no common zero on the closed disc, then there exist func­ tions g^, gg, . . , g^ in H(D, C) such that £^gj+ . . . - 1.

Corollary 3. Every closed ideal in H(D, C) is the principal ideal generated by a function in H(D, C).

Corollary 4. The closed ideals in H(D, C) which are primary, that is, contained in precisely one maximal ideal, are those of the following types;

1. J is the principal ideal generated by (z-a) , where k is a positive integer and a is a point of the open disc.

2. J is the (closed) principal ideal generated by z+X f(z) = (z-X)eP^ ^ , where |X| = 1 and c is a non-negative real number.

There seems to be no end to the applications of ideals!

2. Applications of Ideals in Category Theory

The Additive Category

In fairly recent years, the category has come into prominence as a bona fide algebraic structure. (This structure is thoroughly defined in Appendix A.) Under certain conditions, an ideal of a category can be defined. It is also true that under certain conditions, a category 369 is a ring. In view of these facts, it is inevitable that there would be an accompanying theory of ideals which should certainly include a discussion of radicals. We shall look at those categories with a ring structure first.

An additive category is a category with a ring structure.

DEFINITION VI-34. A category C is an additive category if each

C(A, B) is an abelian group and the composition of morphisms is bilinearJ that is, if we consider

CgCB, C)'C^(A, B) H- Cg(B, C), then if f, geC^(A, B) and hsCgCB, C), then

h(f+g) = hf+hg and if hEC^(A, B) and f, gcCgCB, C), then

(f+g)h = fh+gh.

In view of Definition VI-34, we can observe several remarks due to

Popescu[174].

Remarks

1. The category A of abelian groups is an additive category.

(If X and Y are two abelian groups then the set A^(X, Y) has canonically an abelian group structure: If f, geA^(A, Y), we put f+g:X^Y for the homomorphism of groups defined by

(f+g)(x) = f(x)+g(x).)

2, Let R be a ring with a unit element. Then R is an additive category. (The category A has a single object *, and

R(*, *) = R.

The composition of morphisms is the multiplication of elements of R.

The group structure of R(*, *) is that of the underlying additive group 370

R. Conversely, an additive category C with a single object is a ring;

If X is the object of an additive category C, then the set C(X, X) is a ring frequently denoted by End^(X, X).)

3. Canonically, the category C*, the dual of an additive category C, is also additive. If R is a ring with a unit element, we can denote by R* the dual or the opposite ring of R, that is, the dual category of R.

4. Let C be an additive category. A subcategory C* of C is additive if for any two objects of C', the set C'(X, Y) is a subgroup of C(X, Y). (C* is an additive category if and only if for any f , geC’(X, Y) the difference f-g in C(X, Y) is also in C'(X, Y), If R is a ring, an additive subcategory of R is a subring.)

5. A commutative ring R in which every nonzero monomorphism is an isomorphism (and hence an epimorphism) is called a division ring.

6. A commutative ring R in which every nonzero morphism is an isomorphism is called a field.

The question of what an ideal is in this context still remains to be seen.

DEFINITION VI-35. Let C be an additive category; we call an additive subcategory dP a left ideal of C if satisfies the following condition: If f:X->Y is a morphism of cjP and g:Y^Z is a morphism of C, then gf is a morphism of tif.

Remarks

1. Similarly a right ideal of^can be defined.

2. A two-sided ideal of C is a subcategory of C which is both a right and left ideal of C. 371

In general an Ideal of an additive category may be defined as follows:

DEFINITION VI-36. By an idealj in C is meant the selection for each pair of objects A, B in C, of a subgroup tS)(A, B) of C(A, B) sub­ ject to the requirement that gfhe'iCA, D) whenever geCCC, D), fed? (B, C) and heC(A, B).

Remark

«4^ = ç?(A, A) is an ideal of the ring C^= C(A,A) ,

DEFINITION VI-37. The additive category C is said to be direct if it admits finite direct sums (including the direct sum of no objects,

that is, a null object).

Remarks

1. Any additive category C can be embedded as a full subcategory in a direct category D by taking as an object A of D a finite sequence

A^, Ag, . . , A^ of objects of C and taking D(A, B), where A is as above and B = (B^, Bg, . . ^ B^) to consist of the matrices F = (f^^) where f^jeC(A^, B^).

2. A direct sum of A and B in D is then given by (A^, A^, A^,

* * ’ ^n* ®1* * * ’ ^m^■ DEFINITION VI-38. Let T:C->D be any functor; a congruence & on C called the kernel of T is defined by the following:

f = f'(&) if Tf = Tf'.

Remark Q T then factorizes as C ^ ^ D, where v is injective; and T is

surjective if and only if v is bijactive. 372

According to Kelly[100], if C, D and T are additive, the kernel is an ideal, and s and v are additive. (See Appendix A.)

The Radical of An Additive Category

Now that it has been established that an additive category with one object is a ring and what it means to be an ideal in this kind of ring, we should like to look at the radical of a category.

In 1964, Kelly[100] published some very important results con­ cerning this structure. Some useful lemmas are necessary:

LEMMA 14. If is an ideal of C, then fe<4(P, P') if and only if

f .E^(A , A') for each a,B (see Appendix A). Otp p ot LEMMA 15. If every object of D can be expressed as a finite direct sum of objects of C, then is a one-to-one correspondence between the ideals of D and those of C.

LEMMA 16. If the additive category C is direct, an ideal if of C

is determined by the ideal above. (See Appendix A.)

LEMMA 17. If in a direct category C, we are given for each object A an ideal of the ring C^, then can be extended to an d? of C if and only if gfhe^^ whenever geC(A, B), feA and hcC(B, A). p " LEMMA 18. In a direct category C, let R^ be the radical of

(as a ring) for each object A. Then R^ extends to a unique ideal R of C,

LEMMA 19. If R is the radical of an additive category C, R(A, B)

depends only on the subcategory of C determined by A and B; in fact the necessary and sufficient condition for feC(A, B) to be in R(A, B) is

that 1-gf be a unit of for all gEC(B, A), where R is definedbelow.

DEFINITION VI-39. The radical of C is the trace of R on C. (See

Appendix A.) 373

THEOREM VI-75. R is the greatest ideal of C for which R^ is contained in the radical of C. for each A. A Proof;

For if were any such ideal and feif(A, B), then for any geC(A, B) we should have gfeJ?^ and hence 1-gf would be a unit of C^.

Thus e R.

Since an additive category is actually a ring we should expect more ring-theoretic terminology and notions in the ensuring discussions.

The Brown-McCoy Radical of A Category

In Chapter V we encountered several special ideals called radicals. In particular, one of these radicals was called the McCoy radical in honor of Neal McCoy, an eminent American algebraist.

Gray[68] cites the fact that Neal McCoy, in collaboration with Bailey

Brown, has defined several radicals of rings. Gray provides a very careful analysis of these radicals, one of which is called the

Brown-McCoy radical. We should like to see must what this means in a category setting.

In 1966, Sulinski[193] published a paper which outlined a construction of the Brown-McCoy radical for categories. Some definitions germane to this construction are;

DEFINITION VI-40. Let S be a property of the objects of a

category C. An object A possessing the property S will be called an

S-object. 374

DEFINITION VI-41. An ideal of a category is called an S- ideal if A is an S-object.

DEFINITION VI-42. .If there exists an S-ideal of a category s C a which contains all S-ideals of Ca , then will be called the A g A s Ag

S-radical of Ca . ------*s

DEFINITION VI-43. A category C^ containing no nonzero S-ideals is called S-semi-simpie.

DEFINITION VI-44. Let M be a class of objects of C. We shall say that an object A of C satisfies condition (M) if for each nonzero ideal of C^ there is an epimorphism

f:A^A', f f 0, such that A'EM.

DEFINITION VI-45. A class M will be called regular if each,of its objects satisfies condition (M).

DEFINITION VI-46. An object A is said to be a U^-object.if there is no epimorphism

f:A->B, f # 0 such that Be m ,

DEFINITION VI-47. S is called a radical property if the follow­ ing conditions hold:

1. If A is an S-object and

f:A + B is an epimorphism, then B is also an S-object;

2. For each object A of C, there is an S-radical (A^, R^) of A; 375

3. If

f:A B is such an epimorphism that

(Ag, Rg) = Rer f, then the object B is S-semi-simple.

Remark

If S is a radical property, then the S-objects will be called S- radical objects.

These definitions lead up to the significant result which

Sulinski[193] proved, namely

THEOREM VI-76. If M is a regular class, then is a radical property.

Some further definitions and results of Sulinski are below.

DEFINITION VI-48. A nonzero object A will be called simple if its only ideals are 4(A, 1^) and j^O, 0).

DEFINITION VI-49, Let M be a class of simple objects. An ideal

(A, B) of C(A, B) will be called a simple M-ideal of if BeM.

DEFINITION VI-50. A morphism f of an ideal tf(B, C) of a category

is said to be a retract of A if there exists a morphism g of (A, B) such that

fg = Ig.

DEFINITION VI-51. A class M of simple objects will be called modular if the following conditions hold:

1. If d)(B, C) is a simple M-ideal of C^, then fetf(B, C) is a \ retract of A and there is a unique ideal d?(M, N)eC^ such that

C)ntf(M, N) = (0, 0); 376

2. If «S(B, C) is an ideal of and tJ(D, E) is an M-maximal ideal of Cg, then d?(D, (C, E)) is an ideal of C^.

DEFINITION VI-52. The upper radical property defined by a modular class M will be called the Brown-McCoy radical property.

Remarks

1. In the category of all alternative rings (rings satisfying the property that (ab)b-a(b*b) = 0 = (b*b)a-b(ba)) a class M of simple rings is modular if M contains no simple ring without the unit element.

2. If M consists of all simple rings with unity, then becomes the well-known Brown-McCoy radical.

THEOREM VÏ-77., Let be a Brown-McCoy radical property and let

A be an object of C. Then the ideal tj!(A^, B^) is the U^-radical of A.

THEOREM VI-78. If is a Brown-McCoy radical property, then each U^-semi-simple object can be subdirectly embedded into a direct product of simple objects from M,

THEOREM VI-79., Let be a Brown-McCoy radical property. If

(A, B) is an ideal of a category C^, then

d?(A^, R^B) =d?(A, B)nd?(C^, R^), where i£|(C , R ) is the U -radical of C and tJ(A , R ) is the U radical u u m c u u m of the category C^.

Corollary 1. Each ideal of a U -radical object is a U -ideal. m m Corollary 2. Each ideal of a U^-semi-simple object is also U^- semi-simple.

Though our discussion thus far has been primarily on ideals in additive categories, we know via Definition VI-36 that ideals can be 377 defined for categories whose objects have a ring structure. When this

is done more generalizations of radicals other than the Jacobson radical (discussed earlier) and the Brown-McCoy radical may be attained in a category-theoretic setting. Gray[68] provides an excellent discussion of some of these generalizations of radicals which includes complete analyses of the McCoy and Levitzki radicals among others.

In conclusion, we can assert that ideals form and play a vital role in the foundations of the ring theory that one encounters not only

in algebra but in real analysis, complex analysis, general topology, number theory, category theory and algebraic geometry. In addition, we envision that ideals have applications in non-mathematical disciplines

such as linguistics. We certainly envision other applications of ideals

in the languages usually associated with computer science and artifical

intelligence. APPENDIX A

SELECTED TOPICS FROM ALGEBRA

1. Group Theory

This section gives some of the basic concepts and examples of the group theory used in the body of the paper. The primary references are

Baumslag and Chandler[14], Polites[172], Rotman[179] and van der

Waerden[205].

1. Operation (binary). Let G be a nonempty set; a binary operation on G is a function

f :G'G + G.

2. Group. Let G be a nonempty set of elements and * a binary operation defined on the elements of G. We call G a group with respect to * provided

a. (a*b)*c = a*(b*c) for all a, b, ceG. (Associative

property of *);

b. there is an element e belonging to G such that

a*e = e*a = a

for every aeG (the existence of an identity element);

c. for each aeG, there exists an element yeG such that

a*y = y*a = e

(the existence of inverse elements).

3. Subgroup. A nonempty subset S of a group G is called a subgroup of G provided S is a group under the binary operation of G.

378 379

4. Semigroup. A semigroup is a nonempty set 6 with an associative binary operation.

3. Monoid. A semigroup M with an identity element e is called a monoid.

6. Homomorphism. Let (G, *) and (H, ») be groups with operations

* and ® respectively. A homomorphism f:G-yH is a function for which

f(a*b) = f(a)»f(b) for all a, beG.

7. Isomorphism. An isomorphism is a homomorphism f:G + H, where

G and H are groups,such that f is one-to-one and onto.

8. Automorphism of A Group. An isomorphism f:G->6 of a group G onto itself is called an automorphism.

9. Coset (left). Let G be a group and H a subgroup of G. The set

aH = {ah|heH and aeG, where a is fixed} is called a left coset of H. (A right coset Ha may be defined similarly.)

10. Endomorphism of A Group. A homomorphism f:G-»-G of a group G into Itself is called an endomorphism.

11. Kernel of A Homomorphism. Let f;G + H be a group homomorphism.

The kernel of f is the subset of G,

kernel f = {xeG|f(x) = e}.

12. Group with Operators. Let G be a group and H be a set of elements. G is called a group with operators provided

a. For every 0EÎÎ and for every aeG, 0a is defined and

0aeG; 380

b. For every 0eî2 and a, beG,

0(ab) = 0a»0b.

13. Order of A Group. The order of a group G is the number of

elements in G and is denoted by |G|.

14. Equivalence Relation. A relation R on a set A is called an equivalence relation provided

a. aRa for all aeA (reflexive property);

b. if aRb, then bRa for all a, beA (symmetric property);

c. if aRb and bRa, then aRc for all a, b, ceA (transitive

property).

15. Order of A Group Element, If a is an element of a group G,

then the order of the cyclic group generated by a, denoted by |[a]| , is called the order of a.

16. Some Special Subgroups

a. Normal Subgroup. A subgroup S of a group G is called a

normal subgroup of G provided

aSa ^ = S

for every aeG.

b. Torsion Subgroup. If G is an abelian group, then the

torsion subgroup of G, denoted by tG, is the set of all elements

in G of finite order.

17. Extension of A Group. If K and H are groups, then the group

G is called an extension of K by H provided

a. G contains K as a normal subgroup;

b. ^ - H. 381

18. Isomorphism Theorems for Groups

a. First Isomorphism Theorem. Let f;G->H be a group homo­ morphism with kernel K. Then K is a normal subgroup of G and

I = f(G).

b. Second Isomorphism Theorem. Let S and T be subgroups of

G with T normal in G. Then SPlT is normal in S and

S ST (sriT) " T • c. Third Isomorphism Theorem. Let K c H c G , where both H and H G K are normal subgroups of G, Then — is a normal subgroup of — and

È. G (|) “ « •

19. Some Special Groups

a. Cyclic Group. If aeG, let [a] denote the set of all powers of a. A group G is cyclic if

G = [a] for some aeG.

b. Free Group. Let X be a set and F a group containing X;

F is said to be free on X provided for every group G, there exists a unique extension to a homomorphism of F into G for every function f:X->G; that is,

■ \ X — >G. 382

c. Hamiltonian Group. A group G in which every subgroup is

normal is called a Hamiltonian group.

d. Topological Group. A group G is called a topological

group provided

(1) G is a topological space;

(2) the functions m:G*G->G and u:G-»-G are continuous,

where m is the multiplication (operation) in G and

u(g) = g ^

for all geG.

e. Torsion Group. A group G is called a torsion group if

all of its elements are of finite order.

2. Field Theory

This part of Appendix A gives some of the basic concepts and examples of the field theory used in the body of the paper. The primary references are Artin[4], Birkhoff and MacLane[17], Curtis and Reiner

[29], Dean[31], Kaplansky[98], Lang[129], Polites[172], and van der

Waerden[205].

1. Field. Suppose K is a set containing at least two elements and suppose that there are two binary operations which are denoted by

+ and ‘ defined on the elements of K. We call K a field with respect to + and • if

a. a+b = b+a

for all a, beK (commutativity of +) ;

b. (a+b)+c = a+(b+c)

for all a, b, ceK (associativity of +); 383

c. K contains an element, denoted by 0, such that

a+0 = a

for all asK (0 is called the additive identity element);

d. For each element of K, there is a unique element,

denoted by -a, in K such that

a+(-a) = 0

(-a is called the additive inverse of a);

e. a»b = b*a

for all a, beK (commutativity of •);

f. (a»b)*c = a*(b*c)

for all a, b, ceK (associativity of •);

g. K contains an element denoted by 1 such that

a*l = a

for all aeK (1 is called the unit element, or the multiplicative

identity of K);

h. For each element a f OeK, there is an element denoted by

a"^ and — in K such that a —1 , a*a = 1

(a ^ is called the multiplicative inverse of a);

i. a*(b+c) = a*b+a*c

for all a, b, c e K (distributivity of + over •).

2. Polynomial over A Field. Let K be a field. A polynomial p over K is a sequence (o q , a^, ag, . . . ) of elements from K in which only a finite number of terms are different from 0. Polynomials are usually expressed in the form aQ+a^^x+a^x^+a^s. . . +a^x^, where the a^eK, a^ f 0 if n > 0 and x is an indeterminate. 384

3. Subfield. Let L be a field. A subset K of L is a subfield of

L if K is itself a field relative to the binary operations defined in L.

4. Extension. Let L and K be fields. If K is a subfield of L,

then we call L an extension (field) of K,

3. Degree of An Extension. Suppose L is an extension of a field

K. By the degree of L over K, we mean the dimension of L as a vector

space over the field K which is denoted by [L:K].

6. Adjunction of An Element to A Field. Let L be an extension of a field K and let a be an element of L. Let S be the collection of all subfields of L that contain K and a. The intersection of all the elements of S is denoted by K(a) and a is said to be adjoined to the field K.

7. Integral Element. An element aeL is called integral over K

if there exists a monic polynomial p(x)eK[x] such that

p(a) = 0.

8. Integral Closure, Let K be a subfield of a field L. Then

the elements of L which are integral over K form a subfield of L which is called the integral closure of K in L.

9. Algebraic Element. Let aeL 2 K. a is called algebraic over

K of degree n if a is a zero of an irreducible polynomial p(x)eF[x] of degree n.

10. Separable Element. Let K be a field and let a be an element algebraic over K. If the defining polynomial (irreducible polynomial of smallest degree)

p(x) = 0 has a as a simple root, then a is said to be separable. 385

11. Some Special Extensions

a. Algebraic Extension. An extension L 2 K is called an algebraic extension of K if every element in L is algebraic over

F; that is, every element aeL is a zero of an irreducible polynomial p(x)eK[x] of degree n.

b. Finite Extension. Suppose L 2 K is an extension of a field K. L is called a finite extension of K provided [L:K] is finite.

c. Normal Extension. A finite extension L 2 K is called a normal extension of K provided vdienever an irreducible polynomial p(x)eK[x] has one zero in L, then p(x) splits in L[x]; that is, p(x) has all of its zeros in L.

d. Separable Extension. Suppose L 2 K is an algebraic extension of K. L is said to be a separable extension of K provided every element aeL is separable over K.

e. Simple Extension. An extension L 2 K is called simple if there exists an element aeL such that

L = K(a).

(The element a is called a primitive element of L over K.)

f. Transcendental Extension. Let L 2 K be an extension field of K. If L is not an algebraic extension of K, then it is said to be transcendental.

12. Some Special Fields

a. Algebraically Closed Field. A field L is said to be algebraically closed if every polynomial p(x)eL[x] of degree t 1 has a zero in L. 386

b. Algebraic Function Field. Let C be the field of complex

numbers. Any simple algebraic extension L of the field C[x] is

known as an algebraic function field.

c. Cyclotomie Field. Let 2 be the field of rational numbers

2tt1

0 ) = e = cos (— ) + isin (— ) ; m m then the set

K = Q(w) = {aQ+aj^üH* . . . a^_2* Ia j EQ, 0 < j < M-2}

is a field and, in particular, is called a cyclotomie field. e d. Splitting Field. Let p(x)eK[x], p(x) is said to split

in an extension L of K if p(x) can be factored as a product of

linear factors in L[x]. L is said to be a splitting field for

p(x) over K if p(x) splits in L but in no proper subfield of L

which contains K.

e. Zp (as a field). The set of congruence classes modulo p

is denoted by Zp. Addition and multiplication in 2 are defined by

x + y = x+y, and

X • y = x y , where x is the congruence class of x and when p is a prime number

then Z_ is a field. P

13. Algebraic Closure of a Field. A field L is called an algebraic closure of a field K provided

a. L is an algebraically closed field;

b. L is an algebraic extension of K.

14. Algebraic Independence. Let L 2 K and let

A = (a^^, ag, . . , a^} 387 be a set of n distinct elements in L. The set A is said to be algebraic­

ally independent over K provided

Sk n a. = 0 i j for coefficients k ^ ^ ^ K implies that all coefficients

ki= 0,

15. Galois Group of A Field. If

L = K(a^, ag, . . , a^)

is the splitting field of the polynomial

p(x) = (x-a^)(x-ag) . . . (x-a^),

then the automorphism group of L over K is known as the Galois group of

the equation

p(x) = 0

or the Galois group of the field L over K.

16. Fundamental Theorem of Galois Theory. If G is the Galois

group for the splitting field M of a separable polynomial p(x) over K,

then there exists a one-to-one correspondence

between the subroups H of G and those subfields L of M which contain K.

If L is given, then the corresponding subgroup

H = H(L)

consists of all automorphisms in G which leave each element of L fixed;

if H is given, the corresponding subfield

L = L(H)

consists of all elements in M left invariant by every automorphism of

the subgroup H. For each L, the subgroup H(L) is the Galois group of

M over L and its order is the degree [M:L]. 388

17. Some Special Polynomials

a. Irreducible Polynomial. Let K be a field and p(x) be a

polynomial in K(x), the field of rational functions over K. If

p(x) cannot be written as p(x) = Q(x) r(x)

unless Q(x) or r(x) is a constant, then p(x) is said to be irre­

ducible over K. (It is most important that one specify the field 2 in a discussion of irreducible polynomials; for example, x +5 is

irreducible over Q, the rational field, but is not irreducible

over C, the complex field since

x^+5 = (x+iv'5) (x-i/5). )

b. Minimal Polynomial. Suppose L 2 K is an extension of K.

Let aeK and let p(x) be a polynomial over K of lowest positive degree

satisfied by a. Then p is called a minimal polynomial for a over K.

c. Separable Polynomial. Let K be a field and p(x)eK[x] be

a polynomial. p(x) is said to be a separable polynomial if it

has no multiple roots.

18. Transcendental Element. Let aeL - K. a is called a transcendental element over K if

K(a) - K(x).

19. Valuation of a Field. A field K is said to have a valuation if a function Q(a) is defined for the elements a of K such that

a. Q(a) is an element in an ordered field P;

b. Q(a) > 0 for a f 0; Q(0) = 0;

c. Q(ab) = Q(a)Q(b);

d. Q(a+b) 1 Q(a)+Q(b). 389

3, Category Theory

This part of Appendix A gives some of the basic concepts and examples of the category theory used in some of the material in the body of the paper in an effort to maintain consistency of notions and in notation. The primary references are Gray[68] and Northcott[166].

1. Morphism of a Category. We write f:A B for fehom (A, B) and call f a morphism of a category C with domain A and range B where

A, B are elements of C.

2. Object of a Category. Each element in the class of elements called a category is called an object.

3. Category. Let G be a class of objects (elements) A, B, C

. . . together with a family M of disjoint sets horn (A, B), one for each ordered pair A, B of objects. For each pair of morphisms fehom (A, B), gehom (B, C), there exists a unique morphism gfehom (A, C) defined precisely when range f = domain g. The union of C and M, subject to the following:

a. If fehom (A, B), gehom (B, C), hehom (C, D), then

h(g, f) = (hg)f.

b. For each object B, there exists a morphism lg:B ^ B such

that

1b £ " f,

s Ib = g

always for f : A -*■ B and g:B ■> C.

4. Zero Morphism. The zero morphism 0:A-^B is the unique morphism A-^0 + B, where 0 is the zero object of C* 390

5. Zero Object. A zero object of a category (c is an object with precisely one morphism of Ù to and from any other object.

6. Monomorphism. A morphism v:A B in a category G is said

to be a monomorphism if whenever vu^ = vUg, then = Ug

A — >B

7. Epimorphism. A morphism v:A B in a category G is said to be an epimorphism if whenever u^v = UgV, then = Ug.

C UA J? A — > B

8. Isomorphism. A morphism v:A B in a category G is said to be an isomorphism provided that there exists a morphism u:B A in (?

such that uv = 1. and vu = 1_. A B

A,— >B u

9. Kernel of a Morphism. A kernel of a morphism f:A -- > B of a

category 6 with a zero object is a morphism k:K A such that

a. K — > A — > B = K ~ > B,

b. for any X > A such that

X > A ^— > B “ X ^ > B ,

there is a unique X + K such that

X -- > K — > A = X > A.

10. Subobject. A subject of an object B is an equivalence class

of morphisms into B. 391

Remarks

1. f2 .*^l ® and fg:Ag B of a category C are equivalent if there exist morphisms S^'^l'*’ ^2 ®2*^2^ ^1 that

B and gg B g1 A - A ^ ^ l are commutative; in other words, fgg^ = f^ and fj^gg = fg.

2. The equivalence in Remark 1 is an equivalence relation on the monomorphisme with a fixed object of C a s their range.

11. Normal.-Siibobject. If a subobject is the kernel of some morphism, it is called a normal subobject.

12. Cokernel, The cokernel of a morphism f:A"^B is a morphism u:B C such that

a. A — > B — > C = A — > C

b. for any B*^X such that

A —— > B — > X = A — — > X,

there is a unique C •> X such that

B — > C -- > X = B -- > X.

13. Functor. A functor from a category to a category Cg is an assignment F:C^ to each object of precisely one object of C 2 and to each morphism of precisely one morphism of (g subject to the following:

FI: If 1^ is an identity morphism in C^, then F(l^) is an

identity morphism in Cg.

F2, If fg is defined in C^* then F(f)F(g) is defined in Qg

and 392

F(f)F(g) = F(fg).

If f Is a morphism of then

F(domaln f) = domain F(f)

and

F(range f) = range F(f).

Remarks

1, If and the definition above are small (see Some

Special Categories below), then the functor F is actually a function.

2. The functors defined above are sometimes calledcovariant.

14. Some Special Functors

a. Blfunctor. One can define a functor of two variables,

G : C X C ->■ D given by

G(A^, Ag) = horn (A^, A^)

for any pair of objects A^, A^eC and G(f^, f^):hom (G^, G^)— ^

horn (A|, Ag) is the morphism

G(f^, f^Xy) =

for TCiAjT^Ag, f^:A|-:?A^, ^2*^2— assignment which is a

functor in each of the two variables fj^, f^ separately is called

a bifunctor.

b. Contravariant Functor. A contravariant functor from a

category to a category an assignment F: subject

to the following:

FI. If 1^ is an Identity morphism in then F(l^) is

an identity morphism in

F2. If fg is defined in then F(g)F(f) is defined in

C 2 and 393

F(g)F(f) = F(fg)

is a morphism of

F (domain f) = range F(f)

and

F (range f) = domain F(f).

c. Faithful Functor. A functor "*'^2 defines a collec­ tion of functions horn (A, B) horn (F(A), F(B)) or horn (A, B) -+■ horn (F(B), F(A)); if each of these functions is one-to-one, then

F is called a faithful functor.

d. Forgetful Functor, Let F:A -> S be an assignment which maps any algebraic structure A to its underlying set and maps any map to Itself regarded as a set function; then F is called a forgetful functor.

e. Full Functor. A functor F:defines a collection of functions horn (A, B)horn (F(A), F(B)) or horn (A, B) horn (F(B), F(A)); if each of these functions is onto, then F is called a full functor.

f. Identity Functor. If I, from a category C to itself, is given by

1(A) = A for each object AeC and

1(f) = f for each morphism feC, then I is a functor called the identity functor of C.

g. Inclusion Functor. Suppose C is subcategory (see

Definition 15 below) of G and let J : G' + G be given by 394

J(A) = A

for each object Ae6' and

J(f) = f

for each morphism of C , then J is a functor called the inclusion

functor of in C.

h. Additive Functor. A (covariant) functor F:C + C , where

G and C are additive categories, is additive if for any f, g

raorphisms of G with f+g defined,

F(f+g) = F(f)+F(g);

that is, the induced functions

hom^(A, B) 4. hom^, (F(A) , F(B))

are group homomorphisme.

15. Subcategory. Let G be a category and let G'cG» we say that C* is a subcategory of C if

a. G' is a category;

b. hom^, (A, B) o hom^.(A, B) for all A, BeC.

c. the composition of any two morphisms in t' is the same

as their composition in C.

d. each identity of C is an identity of C.

16. A Special Subcategory: Full subcategory. If in addition to the above definition we have that

hom^, (A, B) = hom^A, B) for all A, Be C* then C is called a full subcategory.

17. Some Special Categories

a. Abelian Category. A category G is abelian if it

satisfies the following axioms: 395

(1) C has a zero object;

(2) Every morphism of G has a kernel;

(3) C is normal;

(4) Every pair of objects of C has a product in t.

b. Additive Category. A category C is additive if

(1) Horn (A, B) is an abelian group for each pair of

objects A, BeC and satisfies the distributive laws

x(f+g) = xf+xg

and

(f+g)x = fx+gx,

whenever both are defined;

(2) G has a zero object.

Remarks

1. Abelian groups constitute an abelian category.

2. An additive category is abelian if it is exact and

has finite products.

c. Dual Category. Let * be a contravariant functor from a category C to a category C* with a contravariant inverse. To each AeCf there corresponds A*eC*» to each morphism f;A BeC, there corresponds f*:B*‘*‘A* in ()*, then we call C* a dual category of C. Note that * is called a duality functor.

d. Normal Category. A category G is called a normal category if all of its objects are normal.

e. Small Category. If G, in the definition of a category above, is a set, then the category G Is called a small category. 396

18, Opposite Ring. In the dual category C* of the category C of all rings, the opposite ring R* is defined as follows: Let ReC. If a, beR, then define

a*b = b*a, where • is the multiplication in R. Hence R* is the ring defined by the operations + and * and is called the opposite ring of R.

19. Direct Sum in Additive Categories. If P is a direct sum of

A^, Ag, . . , A^ in an additive category C, we shall write

p - and shall denote the Injection A^ 4- p by 1^ and the projection

P 4- A by p . Then we have a a p i„ = 6 „ (the Kronecker delta), a B aS

It will be convenient to use

P' = A 1 a for a second direct sum, with maps i*, p'; and so on. a a The maps feG(P, P') are in 1-1 correspondence with the matrices

< V ’ where f _eC(A_, A'); f determines F by a p p ot

and F determines f by

If similarly gE&(P', P") corresponds to the matrix G then gf corres­ ponds to the matrix product GF. 397

Example. Let geC(A, B), heC(B, A). Let P = A @ B, and let

us represent elements of Gp by the corresponding matrices.

We use the fact that x is in the radical of the ring if and only

if 1-yx is a unit for all y in the ring. In this way we see that t 'l-af 0)

-cf 1.

/u 0\ and this has the inverse { j , where u is the inverse of 1-af, \cfu 1/

which exists since fsR.. Then for any eeC_ we have, since R is an A p p ideal, that R contains P

0 0 \ f 0\ 0 h

0 efgh

Thus I \ is a unit of C , and hence 1-egfh is a unit of (J , 0 1-egfh/ P ^

for any eeCg; we conclude that gfhsRg, as required.

4. Ring Theory

This part of Appendix A gives some of the basic concepts and

examples of the ring theory used in some of the material in the body

of the paper in an effort to maintain consistency of notions and in

notations. The primary references are Albert[l], Bames[12], Curtis

and Reiner[29], Gray[68] and Grosswald[73]. 398

1. Ring. A ring R is a nonempty set of elements together with two binary operations, usually designated by + (addition) and * multiplication), subject to the following:

a. R is an abelian group under the operation +, that is,

for all r, seR, r+s is an element of R (closure); for all r,

seR, r+s = s+r (commutativity); there exists an element OeR such

that for each reR, r+0 = r (identity element); for each reR,

there exists an element -reR such that r+(-r) = 0 (inverse

element); for all r, s, teR, r+(s+t) = (r+s)+t (associativity),

b. The operation * is associative; that is, for all r, s

teR, (r*s)*t = r»(s*t).

c. The distributive laws hold for all r, s, teR, r ’(s+t) =

r*s+r*t and (s+t)»r = s*r+t»r.

If further:

d. The multiplication is commutative, that is, r*s = s*t

for all r, seR, then the ring R is said to be commutative.

e. If (a) through (c) are satisfied and there exists an

element leR such that l*r = r»l = r for all reR, R is said to be

a ring with c. unit element.

f. If (a) through (e) are satisfied, then R is called a

commutative ring with a unit element.

2. Subring. A subring S of a ring R is a subgroup of the additive group of the ring R which is closed under multiplication.

3. Ideal. Let R be a ring and let S be a subring of R.

a. S is a left ideal of R if a*x is an element of S for

all elements a of R and x of S; 399

b, S Is a right ideal of R if x*ais an element of S for

all elements a of R and x of S;

c. S is a (two-sided) ideal of R if S is both a left and a

right ideal of R,

Remark

The term ideal is usually used to mean two-sided ideal.

4. Ascending Chain Condition (ACC) on Left Ideals. Let Ij^c Ig c I^ c . . . c= I^cr . . . be a sequence of left ideals of a ring R. If there is an integer N such that

for n > N, then R is said to have the ascending chain condition on left ideals.

Remark

It is also possible to define the ascending chain condition on right ideals and on ideals in a similar manner.

5. Descending Chain Condition (DCC) on Left Ideals. Let

Ij^o 1 ^ 0 l ^ ^ . . . 3 1 ^ 3 . . . be a sequence of left ideals of a ring R. If there is an integer N such that

^n ° ’’N for n > N, then R is said to have the descending chain condition on left ideals.

Remark

It is also possible to define the descending chain condition on right ideals and on ideals in a similar manner. 400

6. Idempotent Element. An element a of a ring R is idempotent

if 2 a = a.

7. Some Special Idempotent Elements

a. Mutually Orthogonal Idempotent Elements. Let e^^, e^,

. . , e^ be nonzero idempotent elements in a ring R. Then the

e^ are said to be mutually orthogonal if

V i - ° whenever i ^ j.

b. Primitive Idempotent Element. An idempotent is primitive

if it cannot be written as the sum of two orthogonal idempotents.

8. Nilpotent Element. Let R be a ring. An element xeR is nilpotent if

x^ = 0

for some positive integer n.

9. Quasi-regular Element. An element aeR is said to be quasi­ regular if there exists an element acR such that

a+a'+aa' = a+a'+a’a = 0.

Remark

One can define both right and left quasi-regular elements in a ring R.

10. Radical of An Ideal. Let a f R be an ideal of a commutative ring R. The radical of a, written /a , is the set of all aeR some positive integral power of which belongs to a. (The set of all elements of R which are nilpotent modulo a.) 401

Example. In the ring of integers, Z, /(O) = (0), /(4) = /(32) =

(2), and / Ô z T = (6).

11. Radical of a Ring. The radical of the ring R is defined to be /(0), the set of all nilpotent elements of R.

Remark

As is indicated in the body of the paper, there are several notions of radicals for various rings.

12. Ring Homomorphism. Let R and S be rings. A function f:R -> S is said to be a ring homomorphism provided

f(a+b) = f(a)+f(b) and

f(a*b) = f(a)*f(b) for all a, beR.

13. Ring Isomorphism. A ring homomorphism f:R S is an isomor­ phism provided there exists a ring homomorphism g:S + R such that

gof - and

fog = Ig.

This is equivalent to saying that f is one-to-one and onto.

14. Ring Isomorphism Theorems.

a. First Isomorphism Theorem. Let R and S be rings and let

f:R-»-s be a ring homomorphism of R onto S. If

a = ker f

then

^ = S. a 402

b. Second Isomorphism Theorem. Let S and T be subrings of

a ring R with S an ideal. Then s H t is an ideal of T, S+T is a

subring of R and

S+I „ T . S " (SflT)

c. Third Isomorphism Theorem. Let S S T S R and S and T T R be ideals of a ring R. Then -g is an ideal of g- and

R S ^ R . T T S

15. Direct Sum. A ring R is the (external) direct sum of rings

R^, Rg, . . , R^ if it is isomorphic to the set of ordered n-tuples of elements of R^, R^, . . , R^ with operations defined pointwlse. Let

R be a ring and let Rj^, R^, . . . R^ be subrings of R. R is the

(internal) direct sum of R^, R^, . . . R^, written

if

R = R,+R_+ . . . +R 1 Z n and for each i = 1, 2, . . . n,

RiH(Ri+R2+ . . . R^_2+ . • . = (0)

Note that this is equivalent to saying that each reR can be written uniquely in the form

r = r^+rj+ . . . +r_^. 403

16. Some Special Ideals

a. Annihilator Ideal. Let S be a nonempty subset of a ring

R. Then

Ap(S) = {xeRjax = 0 for all aeS}

is a right ideal in R, called the right annihilator of S.

Similarly, the left annihilator of a set S can be defined.

b. Contracted Ideal. Let R and S be two rings having

identities and let f:R->-s be a ring homomorphism. If a is an

ideal in the image of f, im f, then

= f ^(o)

is an ideal called the contracted ideal of a.

c. Extended Ideal. Using the hypotheses of (b), above, the ideal

6® = Sf(6) generated by f(B), where B is an ideal of R, is called the extended ideal of B or the extension of B*

d. Fractional Ideal. Let a be a nonzero ideal in the ring of integers R of a quadratic number field F; define a to be the set of elements feF such that faeR for every aeo,' then a is called a fractional ideal.

e. Invertible Ideal. Let R be the ring of integers of a quadratic field F, an ideal aeR is said to be invertible if -1 a a = R.

f. Irreducible Ideal. An ideal a in a ring R is irreducible if whenever 404 a = B Ay, where B and y are Ideals of R, then either

a = B or

a = y.

g. Maximal Ideal. If a is an ideal of a ring R such that a Î* R and for any ideal B such that a c B = R,

3 = a or

8 = R, then a is a maximal ideal of R. (There are similar appropriate definitions for maximal right and maximal left ideals.)

h. Minimal Ideal. If a is an ideal of a ring R such that a f (0) and for any ideal B such that (0) c 3 c a, then

3 = a or

(0) = B, then a is a minimal ideal of R. (There are similar appropriate definitions for minimal right and minimal left ideals of R.)

i. Nil Ideal. An ideal a in a ring R is nil if every element of a is nilpotent.

j . Nilpotent Ideal. An ideal a in a ring R is nilpotent if there is a positive integer n such that

a" = (0), where is the product of a with itself n times. 405

k. Primary Ideal, Let R be a commutative ring with identity.

An ideal a in R is a primary ideal if abea and a^a implyb"ea for

some positive integer n.

1. Prime Ideal. Let R be a commutative ring with identity.

An ideal a of R is said to be a prime ideal of R if a, beR and

abea imply that either aea or bea.

m. Principal Ideal. Let S be a nonempty subset of a ring R.

The left, right or two-sided ideal generated by S is the

smallest left, right or two-sided ideal respectively containing

S, and will be denoted by (S). If S consists of a single

element, say a, then (a) is called the principal (left, right

or two-sided) ideal generated by a.

n. Regular Ideal. A left ideal a of a ring R is called a

regular ideal if there is an element aeR such that R(l-a) c a.

o. Semi-nilpotent Ideal. An ideal a in a ring R is called

semi-nilpotent if each ring generated by a finite set of

elements a^, ag, . . , a^ea is nilpotent.

p. Semi-regular Ideal. An ideal a in a ring R which is not

semi-nilpotent is called semi-regular.

Remarks

1. Nilpotent ideals are semi-nilpotent.

2. Semi-nilpotent ideals are nil.

17. Unit, An element r in a commutative ring R is called a unit if there exists an element s R such that

rs = 1. 406

Remark

In a similar manner one can define left and right units.

18. Zero Divisor. Let R be a commutative ring and let a ^ OeR.

Then a is called a zero divisor if there exists b ^ OeR such that

ab = 0.

19. Integral Domain. A commutative ring R with no zero divisors is called an integral domain. (Some authors prefer to define an integral domain to have a unit element.)

20. Jacobson Radical of a Ring. The Jabobson radical J of a ring

R with identity is a two-sided ideal of R and is equal to the inter­ section of all the maximal left ideals of R.

Remarks

1. The Jacobson radical of a ring R with identity can be equivalently defined to be the intersection of all the maximal right ideals of R.

2. The sum of all the quasi-regular (right, left or two-sided) ideals or R is another characterization of the Jacobson radical,

21. Principal Ideal Domain. A principal ideal domain is an integral domain such that every ideal is principal.

22. Unique Factorization Domain. An integral domain R is a unique factorization domain (UFD) if R has an identity element and if every nonzero nonunit of R can be written as a finite product of prime factors and the factorization is unique to within order and unit factors. 407

23. Module. Let R be a ring; a nonempty set M Is said to be a

left R-module if M is an abelian group under an operation + such

that for every reR and mcM there exists an element rmeM with

a. r(a+b) = ra+rb;

b. r(sa) = (rs)a;

c. (r+s)a = ra+sa

for all a, beM and r, seR.

Remarks

1. Similarly one can define a right R-module.

2. If R has a unit element 1 such that

Im = m

for every meR, then M is called a unital R-module.

24. Module Homomorphism. Given two R-modules M and N of a ring

R, a function f:M ->■ w is called a module homomorphism if

a. f (m^^+mg) = ^ (m^);

b. f(rm) = rf(m)

for all m, m^, mgEM and all reR.

25. Module Series. - A module M is an abelian grouiA hence a normal A series for a module M is a collection of submodules of M

(0) = Mq t= M^ c C . . . C. = M.

26. Submodule. An additive subgroup N of a R-module M is called a submodule 6f M if whenever reR and aeN, then raeN, 408

27. Some Special Modules

a. Completely Reducible Module. A module M is completely reducible if every submodule of M is a direct summand of M.

b. Divisible Module. Let R be a ring. A left R-module M is divisible if

rM = M for all reR.

c. Faithful Module. Let R be a ring. An R-module M is faithful if

rM = (0) for reR implies that

r = 0.

d. Injective Module. A module M is injective if whenever we have

A --- / y t / r

with s injective, then there exists a homomorphism r:B ->• M such that rs = t; that is, every homomorphism from A to M can be extended to a homomorphism from any* module containing A to M.

e. Projective Module. A module M is said to be projective if whenever we have

/ 409

with s surjective, there is a homomorphism r;M -> B such.that

sr = t.

28. Algebra. A vector space A over a field F is said to be an algebra over F if a multiplication is defined which satisfies the following:

a. U'VeA for every u, veA;

b. (u'v)w = u*(v*w) for every u, v, w e A;

c. u*(v4w) = u'V+U'W for every u, v, w e A;

d. (u+v) *w = U'W+V'W for every u, v, w e A;

e. (au)*v = u*(av) = a(u*v) for every aeF and u, veA.

Remark

An algebra, as defined above, was formerly called a hypercomplex system of order n if A, as a vector space, has dimension n.

29. Subalgebra. A subalgebra S of an algebra A is a submodule of A such that the identity of A, l^eS, and s^^, SgsS imply that

®1®2^^ for every s^, s^eS.

30. Some Special Algebras

a. Division Algebra. A algebra D is called a division

algebra if D contains a unit element and if every deD has a

multiplicative inverse in D.

b. Total Matrix Algebra. A total matrix algebra of degree

m over F is the set of all m-rowed square matrices with elements

from F; where F is a field.

31. Some Special Rings

a. Artinlan Ring. A ring R is said to be Artinian or to

have the minimal condition if every nonempty set (P of ideals of 410

R partially ordered by set inclusion has a minimal element; that

is, there is an element ae^ such that if gej? and Bed?, then

a = B.

Remark

One can similarly define right and left Artinlan rings,

b. Dedekind Domain. An integral domain D is a Dedekind

domain if every ideal of D is equal to a finite product of prime

ideals of D.

c. Discrete Valuation Ring. An integral domain D is a

discrete valuation ring if D is a principal ideal domain with

exactly one nonzero prime ideal.

d. Division Ring. A ring R is a division ring, sometimes

called a skew field, if its nonzero elements form a group under multiplication, that is, every nonzero element of R is a unit.

e. Formal Power Series Ring. By a formal power series in n

indeterminates over a commutative ring S with a unit element, one means an infinite sequence

f — (fg, • • • fq, • • * )

of homogeneous polynomials f^ in R, each polynomial fq being

either 0 or of degree q. One defines addition and multiplication

of two power series

and

S (Sgi ®1* ®2* • • • 6q» •

as follows: 411

(1) f+g = (fQ+gQ. • • » fq+gq, • • ■ )

(2) fg (hg, b-|^> hg, • * , hq, « « « )*

where h = Z f g., i+j=q ^ ^ Under these operations, the set R of all formal power series in n indeterminates becomes a ring called the formal power series ring in n indeterminates over S,

f. Matrix Ring. Let M^(R) be the set of all nxn matrices with entries from a ring R. If one defines matrix addition and multiplication in the usual way, then M^(R) is called a matrix ring over R . (See 30 b.)

g. NoetherIan Ring. A ring R is said to be Noetherian if every ideal of R is finitely generated.

Remark

Similarly one defines right and left Noetherian rings.

h. Prime Ring. A ring R is called a prime ring if whenever a and $ are ideals of R and

aB = (0), then either a = (0) or B = (0).

i. Primitive Ring. A ring R is primitive if it has a faithful irreducible module M; that is, M is simple and RM ^ 0.

j. Quotient Ring. Let a be an ideal of a ring R. For a, beR we say that

a = b (mod a) if and only if a-beo.. This defines an equivalence relation on

R. We indicate the set of equivalence classes of R by — and define operations on these disjoint classes by the following: 412

(1 ) (a+a)+(b+a) = (a+b)+a ,

where a+a is the equivalence class of a, where

a+a = {a+b|bea},

(2 ) (a+ct) (b+ct) = ab+a.

— is a ring under the operations (1 ) and (2 ) defined above

and is called the quotient ring of R modulo a.

k. Regular Ring. An element a of a ring R is regular (in the sense of von Neumann[211]) if there exists an element ucR such that

aua = a.

If every element r of a ring R is regular, then R is called a regular ring.

1, Zero Ring, If R is a ring such that the product of any two elements of R is zero we write

= ( 0) and call R a zero ring.

m. Valuation Ring. Let F be a field with a valuation Q.

The set of elements aeF satisfying certain conditions on Q form rings called valuation rings.

n. Simple Ring. If R is not a zero ring and the only ideals of R are (0) and R, then R is called a simple ring.

o. Polynomial Ring. Let R[x] denote the set of all poly­ nomials in an indeterminate x with coefficients in the ring R:

R[x] = {aQ+a^^x+a2 %^+ . . . +a^x^|a^eR, n a nonnegative integer).

If we define addition and multiplication in terms of the 413

operations in R and the laws of exponents, then R[x] is a ring

called the polynomial ring over R, If R is commutative, then

R[x] is commutative; if R has an identity 1, then R[x] has 1 as

its identity.

32. Ideal Class. Suppose a and B are ideals, the ring of integers of K, where K is an algebraic number field, a is equivalent to B, written a - B, if there exist algebraic integers a, bel, the ring of integers in K, such that

(a)a = (b)B.

It can be shown that ~ is an equivalence relation. The classes in­ duced by the equivalence relation " among ideals are called ideal classes.

Remark

The number of ideal classes can be shown to be a finite number.

33. Ideal Class Number. The (finite) number of classes of ideals in an algebraic number field K is called the ideal class number of K and is denoted by

h = h(K).

34. Ideal Operations. Operations involving ideals are defined as follows:

a. Product of Ideals. Let a and B be ideals of a ring R.

The product aB is defined to be the set of all finite sums

n

where the a^ea and the b^sB. For example, in the ring of 414

Integers, If a = (15) and 3 = (12), then

aB = (180).

b. Quotient of Ideals. Let a and B be ideals of a ring R.

The quotient of a and B» written a:B,

a;B = (aeRjabel for all beg}.

Similarly,

B:a = {beRjbasB for all aea}.

For example, in the ring of integers Z, if a = (15) and 6 = (12), then

a:B = (5) and 6 : 0 = (4).

c. Sum of Ideals. Let a and B be ideals of a ring R. The sum of two ideals, written a+B, is given by

a+B = {a+b|aea and bcB).

For example, in the ring of integers Z, if a = (15) and B = (12), then

a+B = (3).

Suppose a, B and y are ideals of the ring of integers I in the algebraic number field K; then the following remarks can be proved:

Remarks

1. The multiplication of ideals is commutative and associative; that is,

aB = By and

(aB)y = a(By). 415

2. An Ideal a has only a finite number of factors; that is,

a - a,a« . . . 1 £ n for some finite number n.

3. Given an ideal there exists an ideal g such that

a-B = (a); that is, such that the product is a principal ideal generated by a rational integer.

4. If (c) is a principal ideal and

(c)a = (c)B, then

a = B.

5. If ya = y3» then a = B-

6. Every ideal a can be factored into prime ideals.

7. If a and B are ideals such that a 3 B, then a is said to divide B, written a|B*

8. If a|B and a f B, then a has fewer factors than B«

9. If a, B and y are ideals in K such that there exists in

K an ideal 6 such that fi|d, 5 |B and y|a, y |B implies that y|S; then 6 is called the greatest common divisor (gcd) of a and

B-

10. If a,B and y are ideals in K such that there exists an ideal 6 such that a|6, B|6 and ajy, B|y implies that ô|y; then

6 is called the least common multiple (1cm) of a and B«

11. A descending chain of ideals

I = I^3p Ig Z) IgZ) . . . 3 1 ^ 3 = (0) 416

Is called a composition series of I if all the quotient ideals

li — are irreducible. 1+1

35. Real Ideal. Let X be a completely regular space and let

M be a maximal ideal in H(X, R). If the quotient field is isomorphic to R, then the ideal M is said to be a real ideal.

Remark H(X R) If — — — fails to be Isomorphic to R and contains R as a proper subfield, then M is said to be a hyper-real ideal.

36. M-n-system. If M is a non-empty m-system, a set N of elements of a ring R is called a right m-system associated with M or briefly a right M-n-system if N contains M and if for every m in M and every n in N there exists an element x of R such that nxmcN.

Remark

If M is the empty set the only right M-n-system is, by defini­ tion, the empty set itself.

37. Contained Module. A module M is said to contain another module M' if every member of M contains M'(Kronecker ' s Modules) .

38. Least Module. In a given finite or infinite set of modules, there is a module which is contained in every other; this particular module is called the least module (of the set of modules).

39. Greatest Module. In a given finite or infinite set of modules, there is a module which contains every other module; this particular module is called the greatest module (of the set of modules). APPENDIX B

SELECTED TOPICS FROM ALGEBRAIC GEOMETRY

This appendix gives some of the basic concepts and examples of the algebraic geometry used in the body of the paper in an effort to maintain consistency of notions and in notation. The primary references are Jenner[96], Lang[130], van der Waerden[205], and Zariski and Samuell222].

1. Point. Let K be a commutative field. A sequence of n elements gg; - ' » « of an arbitrary algebraic extension field L of K is said to be a point of the n-dimensional space

Remark:

The are called the coordinates of the point

g = (%!, Sg, . . . , Q.

2. Affine n-space. An affine n-space over a field K, denoted by A^(K), is the set of n-tuples (x^, Xg, . . . , x^), where for each i.

3. Algebraic Hypersurface. If

f(Xj^, Xg, . . . . x^) = 0 is a polynomial equation in n variables with coefficients in K, the set of points in Ajj(K) satisfying this equation is called an algebraic hypersurface,

4. Algebraic Set. The intersection of a finite number of hypersurfaces is called an algebraic set.

417 418

5. Zero of a Polynomial. Suppose P = x^, . . . ,x^] is a polynomial domain of the n indeterminates %2'' * * its

elements are denoted by

f = f(x) = f(x^, Xg, . . . ,x^)j

a point

Ç - (Gi* ^2..... ^n^

is said to be a zero of the polynomial f(x) if

f(G) = f(Si, ^2' •• • • 'So) “ 6. Algebraic Manifold. The common zeros of an arbitrary number

of polynomials f^^, f^, . , , f^ of a polynomial domain P, that is, the

solutions of a system of equations

fi(ç) = 0. fgCs) = 0,. . . ,f^ce) = 0. are said to form an algebraic manifold.

Remark:

A significant fact concerning albebfaic manifolds^follows:

Let f^, fg, . . . f^ be the polynomials defining the algebraic manifold

M. If the ideal (polynomial)

a — (f^; ^2* *'* * *^n^ is formed, then all points Ç of the manifold (the zeros of f^^, f^, fg,

.. . ,f^) are also zeros of all polynomials

f = * • • "^n^n' where the g^ also belong to the polynomial ideal P as do the f^ then t, manifold M can be defined to be the totality of all the zeros common to

the polynomials of the ideal a, or in particular, of all zeros of the

ideal a. 419

7. Submanifold, An Ideal g which contains a (a divisor of a), that is, a eg, is said to define a submanifold N of a manifold M, where

M is defined in terms of « .

8. Reducible Manifold. A manifold which may be represented as the union of two proper submanifolds is said to be reducible or composite.

9. Irreducible Manifold. A non-composite or a non-reducible is said to be indecomposible or irreducible.

10. Variety. A algebraic set V is called a variety if it cannot be expressed as a proper union of two algebraic sets; that is, if

V f AuB, where A and B are algebraic sets distinct from V.

11. Some Special Varieties:

a. Algebraic Variety. An algebraic variety in an affine

space Ay,(K) is an subset of \ (K) which is the variety of some

ideal inLfxjjX^, ... , where L is a subfield of the

algebraically closed field k.

b. Reducible Variety. A variety V defined over a field K is

reducible over K if it can be decomposed into a sum of two

varieties V^ and V^ which are defined over K and are proper

subsets of V.

12. Parametric Representation. If gg* ■• • algebraic functions of t^, t^, . . . ,t^, the function values .... ,5^ belonging to the special allowable argument values T2 * • • • determine a point Ç' in R^; then we say that we have a parametric 420 representation of the manifold by the algebraic functions ......

of the parameters tj^, ...... t^.

13. Dimension of a Manifold. The smallest number of parameters t^, tg, . . . ,t^ chosen from the field K(g^, . . . ,C^), where

^1* ^2’ ** ‘ *^n *:he algebraic functions of t^, tg, . . . ,t^ , required to form the parametric representation of a manifold M is called the dimension of the manifold M (sometimes called the dimension of the prime ideal p belonging to M).

14. Curve. Suppose A^(K) is an affine n-space over a field K.

If H is an algebraic hypersurface defined over K and n = 2, then H is called a curve.

15. Surface. Suppose A^(K) is an affine n-space over a field K.

If H is an algebraic hypersurface defined over K and n = 3, then H is called a surface.

16. Hyperplane. A hypersurface given by an equation of degree 1 is called a hyperplane in (k)

17. Line. A hypersurface given by an equation of degree 1 is called a line in (k)

18. Plane. A hypersurface given by an equation of degree 1 is called a plane in A^(k)

19. Some Special Points:

a. Generic Point. A point

C “ (S^* Ggi* • •

is a generic point of the manifold M if every algebraic equation 421 with coefficients in a field K, which is valid for g is valid for all points of M and conversely.

b. Multiple Singular Point (x-fold), A point (x^, y^) on a curve is called a multiple singular point (t-fold) if all partial derivatives of order, strictly less than t , vanish at ( x ^ , y^) but some T-th partial derivative does not vanish at (x^, 0^).

c. Branch Point. A branch point is a point on the complex plane where a multiple-valued function w(z) has the double property that

(1) two or more values of w become equal

(2) as z describes once a small circle with center z ' o w does not describe a closed curve about w . o APPENDIX C

SELECTED TOPICS FROM ANALYSIS

This appendix represents some of the basic concepts and examples of the complex analysis, functional analysis and real analysis used in some of the material in the body of the paper in an effort to maintain consistency of notation and notions. The primary references are

Davis[30], Hille[87], Royden[180] and Taylor[194].

1. Sequence. A function whose domain is a subset of the natural numbers is called a sequence,

2. A Special Sequence; Cauchy Sequence. A sequence (x^) of real numbers is called a Cauchy Sequence if given e>0, there is an N such that for all n ^ N and all ro ^ N, we have that

3. Series. If a^, a^, . . . a^, . . . is a sequence of numbers, the expansion

E a = a,+a„+ . . . +a + . . . n^l n 1 2 n is called an infinite series or briefly a series.

4. A Special Series: Power Series. Suppose a^, ag, . . . a^,

. . . is a given sequence of complex numbers; then the series

E a z" = a„+a,z+a„z^+ ...» n=0 u ± z where z is a complex number, is called a power series.

422 423

5. Limit of a Sequence. A real number A is a limit of a sequence

if for each positive e, there is an N(e) such that for all n ^ N, we have that

6. Metric. A real-valued function cr:XxX-^R such that for all

X , y and z in X, we have

a. o(x, y) > 0;

b. a(x, y) = 0 if and only if x = y;

c. o(x, y) = a(y, x) ;

d. o(x, y) < o(x, z)+a(z, y) is called a metric.

7. Metric Space. A set X with a metric a defined on it is called a metric space.

8. Norm. Let X be a linear (vector) space. A nonnegative real­ valued function || [|:X-+R is called a norm if

a. ]|x|| = 0 ■«-»' X = e, the zero vector;

b. ||x+y|| â l|xl| + ||yl|;

c. ||ax|| = |a| ||x|| .

9. Complete Space. A normed linear space is called complete if every Cauchy sequence in the space converges; that is, if for every

Cauchy sequence ^f^) in the space, there is an element f in the space such that

10. Banach Space. A complete normed linear space is called a

Banach space. 424

11. Isometric Spaces. Two metric spaces X and Y are said to be isometric if there is a function f:X-»Y such that f~^;Y+X exists and further

d(x^, x^) = d(f(x^),f(x^)) for every pair of points x^, XgCX, where d is the metric of X.

12. Completion of a Normed Linear Space. Let X be an incomplete normed linear space. There exists a complete normed linear space Y and a subset Y q dense in Y such that Y^ and X are isometric. The space Y is called the completion of X.

13. Denseness. A set D c R is said to be dense in R if

D = R, where D is the union of D and its limit points, D ’, that is,

D = DUD'.

14. Perfect Set. A set S is called perfect if every point of S is a limit point; that is,

S' = S.

15. Measure. Let E be a set. The measure of E, denoted m(E), is a function defined on E satisfying the following conditions:

a. m(E) is defined for each set E;

b. m(E) > 0;

c. If

E = Û E , i=l

where is anyi sequence of sets, then

m(E)<= 2 m(E.); i=l ^ 425

d. If

E = ü E., 1=1

where the E^^ are mutually disjoint, then

m(E) = E m(E.); i=l ^ e. If E^^ c Eg, then m(E^) < m(Eg);

f. If E is an interval, then

m(E) = A(E),

where A(E) is the length of the interval.

g. If E is a set for which m(E) is defined and if E+y is the

set {x+y|xeE} obtained by replacing each point x in E by the

point x+y, then

m(E+y) = m(E).

16. Pole. Suppose that f can be expressed by

f(z) = .y + . . . + a-1 + I ( z - a f , (z-a) (z-a) (z~a) n=0 where the a_^, z and a are complex numbers. The point z = a is called a pole of order m.

17. Riemmann Sphere. In a three-dimensional Euclidean space with coordinates (x, y, z) we identify the (x, y)-plane with the complex plane C; the sphere

x^+y^+z^ = u^ is sometimes called the Riemann sphere.

18. Some Special Algebras of Analysis:

a. Normed Algebra. An algebra which is also a normed linear

space is called a normed algebra. 426

b. Banach Algebra. A normed algebra which is also a Banach

space is called a Banach algebra.

19. Order of Magnitude. The notation

f = 0(g) or equivalently

f(x) = 0(g(x)) as X approaches some limit (finite or infinite) means that within some deleted neighborhood of that limit, f(x) is dominated by some positive constant multiple of g(x); that is,

|f(x)I 5 K*g(x).

If simultaneously, f = 0(g) and g = 0(f), the two functions f and g are said to be of the same order of magnitude as x approaches its limit.

20. Poisson Integral Formula. The formula

u(r, *) - i ^ R -2Rrcos(0-(fi)+r i0 i6 where z = Re and Zq = re , (r < R) are equations of circles, is known as the Poisson Integral Formula.

21. Some Special Functions;

a, Holoraorphic Function. A function f is differentiable in

D if it is differentiable at points of D; f(z) is then said to

be a holomorphic function of z in D. f(z) is said to be holo-

morphic at z = Zq if it is holomorphic at some e-neighborhood of

Zq .

b. Entire Function. A function holomorphic in the finite

complex plane is called an entire function. 427

c. Transcendental Function. An entire function which cannot

be expressed as a polynomial function is called a transcendental

(entire) function,

d. Integral Function. The British name for any transcendental

entire function,

e. Meromorphic Function. A function f is said to be

meromorphic in a domain D if it has no singularities other than

poles in D (see 22 below).

f. Rational Function. A rational function h is a function

that can be written as a quotient

where f and g are polynomial functions,

22. Singular Point. A point at which a function f(z) fails to be holomorphic is called a singular point or a singularity of f(z)> where z is a complex number.

23. Cantor Set. Consider the unit interval [0, 1]. Remove from 1 2 1 2 it the open middle third (^ , ^). From [0, y] U [-j, 1] remove the

1 2 7 8 middle thirds ( ^ , -g), (g-, g- ) • Continue in this manner an infinite number of steps. What remains of [0, 1] is known as the Cantor set, and is usually denoted by C. Some of the remarkable properties of this Interesting set are as follows:

a. The Cantor set is non-denumerable.

DISCUSSION: Every number in the Cantor set has a ternary

expansion, that is, every number in this set is of the form 428

where each a^ is either 0 or 2. Also each number in [0, 1] has a binary expansion, that is, an expansion of the form

i=l 2^ ^ 2^ where each a^ is either 0 or 1 and we assume that not all the b^ after a certain term are 0. Then one can establish a 1-1 correspondence between the numbers in the binary and ternary forms such that b^ = 0 when a^ = 0 and b^ = 1 when a^ = 2.

Then since the set of all real numbers in [0, 1] is non- denumerable, then the Cantor set is non-denumerable. b. The Cantor set is measurable and has measure 0.

DISCUSSION: As is indicated above in 23, the Cantor set is obtained by taking the interval [0, 1] subdividing it into three parts and removing the middle third. Then the remain­ ing parts are divided into thirds and their middle thirds are removed. This process is repeated an infinite number of times and what is left is called the Cantor set. During the first step, the interval of length 1/3 is removed. During the second step, two intervals of lengths — are removed. It 3 follows then that the total measure of the Cantor set C is

, 1 1 ? 2”-^ _ , 3 ______, 1 M(C> = 1- — — = 1- — Ô = 1-1 - 0, 1 - = 1-1, 3 " 1 - y 1~ ~ where one uses the formula for the sum of a geometric series. 429

c. The Cantor set is a perfect set.

DISCUSSION: The Cantor set C is the complement relative

to [0, 1] of an open set (%(C), where

C(C) = (^, 0 (^, “ ) V (^, ^) . . . .

Hence C is closed. Every number in C is a limit point of

numbers having terminating ternary expansions. But aUr the

numbers having terminating ternary expansions all belong to

C, Hence C is dense-in-itself, Since C is closed and

dense-in-itself, C is perfect.

24. Boundary-value Problem. A problem of finding a solution to a given differential equation or set of equations which will meet certain specified requirements for a given set of values of the independent variables, sometimes called the boundary points is called a boundary value problem.

25. Rational Function, A function f(z) which is equal to the quotient of two polynomials

■ b”+b^z+ . . . +bV •’n whenever'the denominator is different from zero, and equal to the limit of this quotient at every point z = a or z = “ , where this limit is finite and z e C» is called a rational function.

26. Divisor of a Rational Function. If the pol^s and zeros of a rational function f(x, y) are at the cycles Ug, . . . u^, and have the orders pg, . . . then the divisor

^ “f = “l “ 2 • • • “s is called a divisor of f. CSee Appendix C.) APPENDIX D

SELECTED TOPICS FROM LINEAR ALGEBRA

This appendix represents some of the basic concepts and examples of the linear algebra used in some of the material in the body of the paper in an effort to maintain consistency of notions and in notation.

The primary references are Albert[1, 2] and Nering[161].

1. Spanning Set. For any subset A of V, a vector space, the set of all linear combinations of vectors in A is the set spanned by

A. :

2. Linearly Dependent. A set of vectors of a vector space V is said to be linearly dependent if there exists a non-trivial linear relation among them; that is, for the set of vectors a^, a^, . . . a^ there exist scalars c^ not all equal to zero such that

c,a-+c-a^+ . . +c. = 0. 11 2 2 - n n 3. Linearly Independent. The vectors a^, a^, . . . are linearly independent over a field F if and only if for all scalars c^eF

c,a,+c„a„+ . . . +c a = 0 1 1 z z n n implies that

4. Basis. A linearly independent set spanning a vector space V is called a basis.

430 431

5. Dimension. The number of elements in a basis is called the dimension of the vector space.

6. Matrix. A matrix over a field F is a rectangular array of scalars a^eF,

7. Homogeneous Polynomial. A polynomial

f(x^, Xg, . . . x^) = 0 is called a homogeneous polynomial if the degrees of all of its terms are equal.

8. Characteristic Matrix. The matrix tl^-A, where is the n-square identity matrix, A is an n-square matrix over a field f, and t is an indeterminate is called the characteristic matrix of A.

9. Characteristic Polynomial. Let

C = tl -A n be the characteristic matrix of A. The determinant of C is a polynomial

|c| = |tI^-A| is called the characteristiq*polynomial of A.

Remark

The equation |c| = |tI^-A| = 0 is called the characteristic equation of A.

10. Trace. Let A be an nxn matrix. The trace of A is defined to be the sum of the elements in the main diagonal of A, that is, APPENDIX E

SELECTED TOPICS FROM NUMBER THEORY

This appendix gives some of the basic concepts and examples of the number theory used in the body of the paper in an effort to maintain consistency in notation and of notions. The primary references are

Griffin[72], Grosswald[73] and Ireland and Rosen[92].

1. Congruence (in Z). If a, b, raeZ and m 0, we say that a is congruent to b modulo m if m divides b-a. This relation is written

a = b(mod m) ox

a = b(m) .

Remark

Congruence modulo m is an equivalence relation on the set of integers,

2. Belonging to an Exponent. Suppose a and m are integers. Then if d is the least positive integer such that

a*^ = 1 (mod m) ; d is called the exponent to which a belongs modulo m; or we say that a belongs to d modulo m.

3. Crelle's Table. In 1844, A. L. Crelle gave a device for finding the exponent to which an integer a belongs modulo m. To employ this method one first sets up the integers 1, 2, . . . m-1 in a row and under 1 one puts r^, the least positive residue modulo m of the

432 433

Integers a; under 2 one puts the least positive residue r^ of

(rj^+a) = 2r^^ (modulo m) ;

under 3 put the least positive residue of

(rg+a) = rg+r^(mod m).

Then the resulting table

1 2 3 4 m-1 H '^2 >^3 ^ ’^m-1 gives in order, in the second row, the residues of

r^, 2r^, 3r^, . . . (m-l)r^modulo m.

According to this scheme, the integer r^ congruent to a modulo m is

under 1; the integer congruent to 2 a = r^«rg(mod m)

is under r^ and likewise the integer congruent to

a® = t*r^(mod m)

is under t. For example, if m = 7 and we wish to find the exponent to

which 3 belongs mod 7, we have in the table below

1 2 3 4 5 6 3 6 2 5 1 4.

But

3 = 3(mod 7)

so we move to 3 in the first row. We find 2 under 3 and hence

3^ = 2(mod 7).

Continuing we find that

3^ = l(mod 7).

Hence 3 belongs to 6 modulo 7. 434

4. Algebraic Number Field. An algebraic number field is a field of complex numbers which is a finite-dimensional vector space over the field of rational numbers.

5. Algebraic Integer. An algebraic integer w Is a complex number which is a root of a polynomial equation

x'^+bj^x" . . . +b^ = 0, where b^, bg, . . . b^ez. (Note that a rational number reQ is an algebraic integer if and only if rez.)

6. Conjugate. Let

f(x) = x"fa^^x" ^+agX^ . . . +a^, where the a^EQ, be irreducible over Q. The n roots of

f(x) = 0 are all distinct and will be denoted by

e =

The algebraic numbers not necessarily integers, are called the conjugates of 6.

7. Norm. Let cx^, . . . be the conjugates of the algebraic integer a. The norm of ot, written Not, is the product of

. . . , that is.

Not = .

8. Unit. otez(0) is called a unit if

Not = 1 , where Z(9) is the ring of integers of an algebraic number field Q(0).

9. Diophantine Equation. A diophantine equation is an equation with integer coefficients which requires integer solutions. 435

10. Greatest Common Divisor. Let a,beZ. An integer d is called a greatest common divisor of a and b if d is a divisor of both a and b and if every other common divisor of a and b divides d.

11. Least Common Multiple. A least common multiple of two or more integers is the least integer that is divisible by the given integers.

12. Pell's Equation (The Pellian Equation). The equation

x^-by^ = 1, in which b is a positive integer which is not a square, is known as

Pell's equation.

13. Primitive Root, a is a primitive root mod p if p-1 is the smallest positive integer such that

aP“^ 5 l(p).

14. Gaussian Integers. The ring Z(i] is called the ring of

Gaussian integers.

15. Trace, Let aeK^E, the trace of a from E to K, where K is an extension field of E, is defined by 2 qh-1 tr(a) = a+a + . . . +a^

16. Euler-(|>-function. For nez'*', <|)(n) is definedj:o be the number of positive integers

*(1) = 1, <|i(5) = 4, 4>(6) = 2, and ((,(9) = 6 . If p is a prime, it is clear that

*(p) = p-1*

17. Primary Prime. If it is a prime in Z[m], where m is a primitive cube root of unity, we say that it is primary if

IT = 2 (mod 3). 436

18. Quadratic Residue. Let (r, m) = 1 ; then r is said to be a quadratic residue modulo m if there exists,some integer x such that 2 X = r(mod m),

19. Quadratic Reciprocity Law. If p and q are distinct odd primes, then

(^> (f ) - ( - 1 ) ' ^ ” ^ ’ . where (^) and (^) are Legendre symbols.

20. Cubic Residue Character. If Nr f 3, where r is a prime in

Z[üi], then the cubic residue character of a modulo r is given by

a. (f-)^ = = 0 if r|aj

(Ntt-1) 9 3 a 2 b. a = x^(a)(Ti), with ( ^ > 2 equal to 1, o), m .

Remark

The character plays the same role in the theory of cubic residues as the Legendre symbol plays in the theory of quadratic residues,

21. Cubic Reciprocity Law. Let and Vg be primary primes in

Z[w], NtTj^ 3, NVg 4 3 and Nr^ ^ NWg. Then

22. Logarithmic Determinant. The logarithmic determinant is defined to be the determinant

L Cj^(a), L Cg(a) , . . . L^_^(a)

L C^(a^), L Cg(a^), . . . LC .(a^)

p-2 p-2 p—2 L C^(a^ ), L CgCa"'^ ), . . . L C (a^ ) 437 where C^^(a), Cg(a), . . . ^ (a) is a system of independent units, y is a primitive root of p different from zero and L C^(a), L Cg(a), . . are the arithmetical logarithms of the real units C^(a) taken positively. APPENDIX F

SELECTED TOPICS FROM TOPOLOGY

This appendix gives briefly some of the basic concepts and examples of the number theory used in the body of the paper in an effort to maintain consistency of notions and in notation. The primary references are Dugundji[44], Kelley[101], and Maunder[149].

1. Topology. Let X be a set. A topology (or topological structure) in X is a family 3 " o f subsets of X that satisfies

a. Each union of members of 3" is, also a member of

b. Each finite intersection of members of T is also a

member of y;

c. 3" and 4 » are members of T.

2. Topological Space. A couple (X, 3^ consisting of a set X and a topology 3"is called a topological space.

Remark

The elements of X are called open sets.

3. Closed Set. A set A c X is called a closed set if the complement of A in X, or X-A, is open.

4. F -set. A set F is called an F -set if it is the union of o a at most countably many closed sets.

5. G --set. A set G is called a G --set if it is the intersection 0 0 of at most countable many open sets.

438 439

6. Relative Topology. Let ( X , '3') be a topological space and

YcX. The relative (induced) topologyon Y is {YnujeT}» (Y, J'y) is called a subspace of (X, T ) •

7. Discrete Topology. Let X be a set and P(X) be the set of all subsets of X.

T = P(x) is called the discrete topology and every set is an open set in X*

For example, let

X = {0. 1}; then

T = {*, X, {1}, {0}} is the discrete topology on X.

8. Indiscrete Topology. Let X be any set. Then

T = {<{>» X} is a topology such that 4> and X are the only open sets and is called the indiscrete topology. For example; if

X = R, the real line, then

T ~ {* X } is the indiscrete topology.

9. Neighborhood. Let (X, X) be a topological space. By a neighborhood (nbd of xeX) is meant any open set; that is, any member of

T containing x,

10. Basis (for a topology). Let (X,T) be a topological space.

A family BcJ*i8 called a basis for X if each open set, that is, each member of X» is the union of members of B. 440

11. Closure. Let AcX. A point xgX is adherent to A if each neighborhood of x contains at least one point of A (which may be x itself). The set

A = {xex|v U (x) : U (x) n A 4») of all points in X adherent to A is called the closure of A.

12. Interior. Let Acrx. The interior, Int(A), of A is the largest open set contained in A,

13. Nowhere Dense. A set A in a topological space X is nowhere dense In X if

Int(A) = 4>.

" ■; ;■ ■ 14. Character of a Topological Space. The character of a topological space T i s the least cardinal number belonging to a basis for X*

15. Open Map. A map f:X*^Y is called open if the image of each set open in X is open in Y, where X and Y are topological spaces.

16. Continuous Map, A map f:X-*-Y is called continuous if the inverse image of each set open in Y is open in X, where X and Y are topological spaces. K Remark

Let X and Y be topological spaces, and ftX'^’Y a map. The following statements are equivalent:

a. f is continuous.

b. The inverse image of each closed set in Y is closed in X. 441

à.t For each xeX and each neighborhood W(f(x)) in Y, there

exists a neighborhood V(x) such that f(V(x))cW(f(x)).

d. f(A) (= f(A) for every A c X.

e . f”^(B) c f"^(B) for every BcY.

17. Homeomorphism. A continuous bijective map f:X4-Y such that f ^:Y + X is also continuous, is called a homeomorphism (or a _ f bicontinuous bijection) and is denoted by X ^ Y„

18. Covering. Let Y be any set, thenif{A^|aeA} is a . family of ;

subsets of X such that

it is called a covering of X.

19. Some Special Topological Spaces

a. Hausdorff Topological Space. A topological space Y is

called a Hausdorff space if each two distinct points have non-

intersecting neighborhoods; that is, whenever p f q there are

neighborhoods U(p), V(q) such that

U(p) nv(q) = (|).

b. Compact Topological Space, A Hausdorff space Y is

compact if each open covering has a finite subcovering.

c. Connected Topological Space. A space X is connected if

it is not the union of two nonempty disjoint open sets.

d. Locally Compact Topological Space. A Hausdorff spaceis

locally compact if each point has a relatively compact neighbor­

hood U, that is, U is compact.

e. Regular Topological Space. A Hausdorff space is regular

if each yeY and closed set A not containing y have disjoint 442

nelghboorhoods that is, if A is a closed set and y d A, then

there is a neighborhood U of y and an open set V ^ A such that

vnu =

f. Completely Regular Topological Space. A Hausdorff space

y is called a completely regular space or a Tyconoff space if,

for each point peY and closed set A not containing p, there is

a continuous map f:Y + I = [0, 1] such that

f(p) = 1

and

f(a) = 0

for each aeA.

g. Totally Disconnected Topological Space. Suppose Y is a

topological space and yeY. The component of y, C(y), is defined

to be the union of all connected subsets of Y containing y. Y

is called a totally disconnected space if

C(y) = y.

h. Metrizable Topological Space. A topological space (Y, 31

is called a metrizable space if its topology is induced by a

metric in Y. A metric for a space Y is one that induces a

topology. (Metric was defined in Appendix C.)

20. Uniform Convergence on Compact Sets. Let (Z, d) be a metric space and Y an arbitrary space. A sequence {f^} of continuous functions from Y into Z is said to converge to a function f:y + z uniformly on every compact subset if for every compact C c Y and each e > 0, there is an integer N = N(C, e ) such that 443

d(f(c), f^(c)) < E for every n ^ N and every ceC.

21. Preorder. A binary relation R in a set A is called a preorder if it is reflexive and transitive; that is, if

a, VaiaRa

b. (aRb) A (bRc) -> aRc.

Remark

A set together with a definite preorder is called a preordered set.

22. Partial Ordering. A preordering in a set A with the additional property that

(a < b) A (b < a) (a = b) is called a partial ordering.

Remark

A set together with a definite partial ordering is called a partially ordered set.

23. Chain. Let A be a preordered set with the relation <. B c A is called a chain in A if each two elements of B are related.

24. Total Ordering. Let A be a set. A total ordering on A is a partial ordering < such.that, given a, bsA, then either a < b or b < a.

25. Upper Bound (for a subset B e A). a^sA is called an upper bound for a subset B<=A if VbEB:b < a^, where A is a preordered set. 444

26. Maximal Element. Let A be a preordered set. msA is called a maximal element in A if—Va:m■ a < m; that is, if either no aeA follows m or each a that follows m also precedes m.

27. The set Z(X); Let X be a completely regular space. Z(X) is defined to be the family of the form z(f); where f is an element of

H(X, R). If M is any subset of H(X, R), then Z(M) is defined to be the family of all z(f) for feM. (Symbols defined in Chapter VI).

Remark

For various special categories of spaces, the family Z assumes familiar forms. If M is a metric space, for example, Z(M) coincides with the family of all closed subsets of M,

28. Theorem (Urysohn). Let Y be a Hausdorff space. The follow­ ing two properties are equivalent;

a. Y is normal. (Hausdorff such that disjoint closed sets have disjoint nbds). b. For each pair of disjoint sets A and B in Y, there

exists a continuous function f:Y->R, called a Urysohn function

for A and B, such that:

(1) 0 £ f(y) < 1 for all yeY.

(2) f(a) = 0 for all aeA.

(3) f(b) = 1 for all bsB.

29. Quotient Set. Let A have an equivalence relation R. The set whose elements are the R-equivalence classes is called the quotient set of A by R and is written V r . The map p^rA'^ V r given by a->Ra is called the projection of A onto ^/r .

V r . 4^5

30. Identification Topology. Let Y be an arbitrary set, X a topological space and p:X Y a surjection. The identification topology in Y determined by p is

y p ) = {UCY| p “^(U) is open in X}.

31. Quotient Space. The set with the identification topology determined by the projection p;X-> ^/r is called the quotient space of

X by R.

32. Theorem (Stone-Cech).

a. For each compact space Y and each continuous function

f:X ->Y, there exists a unique continuous function F: gX+ Y

such that

f = F ■> p,

where p:X+P^, the cartesian product of closed unit intervals

and 3X is a compact Hausdorff space such that p is homeomorphism

of X onto a dense subset of 3X. A b. (Uniqueness). Any compactification (X, h) of X (see 33

below) having the property (a) above is homeomorphic to gX;

indeed there is a homeomorphism,

X = ex,

that is the identity map on X.

A c. ex is the "largest" compactification of X: if X is any

compactification of X, then X is the quotient space of gX.

33. Compactification of â Space X. A compactification of a

A A space X is a pair (X, h) consisting of a compact Hausdorff space X and

A a homeomorphism h of X onto a dense subset of X. 446

34. The Stone-Cech Compactification of X. The Stone-Cech compactification of X is the pair (gX, p), where p is an embedding p:X+p^ (see 32 above),

ex = P(X), and has the properties of the Stone-Cech Theorem, BIBLIOGRAPHY

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