Rings and Ideals

AMS / MAA THE CARUS MATHEMATICAL MONOGRAPHS VOL 8

Rings and Ideals

Neil H. McCoy 10.1090/car/008

RINGS AND IDEALS

By NEAL HENRY McCOY Professor of Mathematics, Smith College

THE CARUS MATHEMATICAL MONOGRAPHS

Published by THE MATHEMATICAL ASSOCIATION OF AMERICA

Editorial Committee PHILIP FRANKLIN, Chairman A. J. KEMPNER GARRETT BIRKHOFF E. F. BECKENBACH J. L. SYNGE H. S. M. COXETER HE CARUS MATHEMATICAL MONOGRAPHS are an expres- sion of the desire of Mrs. Mary Hegeler Carus, and of her son, Dr. Edward H. Carus, to contribute to the dissemination of Tmathematical knowledge by making accessible at nominal cost a series of expository presentations of the best thoughts and keenest researches in pure and applied mathematics. The publication of the first four of these monographs was made possible by a notable gift to the Mathematical Association of America by Mrs. Carus as sole trustee of the Edward C. Hegeler Trust Fund. The sales from these have resulted in the Carus Monograph Fund, and the Mathematical Association has used this as a revolving book fund to publish the succeeding monographs. The expositions of the mathematical subjects which the monographs contain are set forth in a manner comprehensible not only to teachers and students specializing in mathematics, but also to scientific workers in other fields, and especially to the wide circle of thoughtful people who, having a moderate acquaintance with elementary mathematics, wish to extend their knowledge without prolonged and critical study of the mathematical journals and treatises. The scope of this series includes also historical and biographical monographs.

The following books in this series have been published to date:

No. 1. Calculus of Variations, by G. A. Bliss No. 2. Analytic Functions of a Complex Variable, by D. R. Curtiss No. 3. Mathematical Statistics, by H. L. Rietz No. 4. Projective Geometry, by J. W. Young No. 5. A History of Mathematics in America before 1900, by D. E. Smith and Jekuthiel Ginsburg No. 6. Fourier Series and Orthogonal Polynomials, by Dunham No. 7. Vectors and Matrices, by C. C. MacDuffee No. 8. Rings and Ideals, by Ν. H. McCoy No. 9. The Theory of Algebraic Numbers by Harry Pollard No. 10. The Arithmetic Theory of Quadratic Forms by B. W. Jones No. 11. Irrational Numbers by Ivan Niven No. 12. Statistical Independence in Probability, Analysis and Number Theory by Mark Kac No. 13. A Primer of Real Functions by Ralph P. Boas, Jr. No. 14. Combinatorial Mathematics by H. J. Ryser No. 15. Non-Commutative Rings by I. N. Herstein The Car us Mathematical Monographs

NUMBER EIGHT

RINGS AND IDEALS

NEAL H. McCOY Professor of Mathematics Smith College

Paperback ISBN 978-0-88385-137-1 eISBN 978-1-61444-008-6 Hardcover (out of print) ISBN 978-0-88385-008-4

Printed in the United States of America

Current printing (last digit): 10 9 8 7 6 5 PREFACE During the last twenty or thirty years abstract algebra has been developed in very rapid fashion by an increas- ingly large number of research workers. In fact, the general methods and most fundamental results of the theory have become of considerable interest to mathematicians gen- erally, even though their primary interests may lie in other directions. The purpose of this monograph is to present an introduction to that branch of abstract algebra having to do with the theory of rings, with some emphasis on the role of ideals in the theory. Except for a knowledge of certain fundamental theorems about determinants which is assumed in Chapter VIII, and at one point in Chapter VII, the book is almost entirely self-contained. Of course, the reader must have a certain amount of "mathematical maturity" in order to understand the illustrative examples, and also to grasp the significance of the abstract approach. However, in so far as formal technique is concerned, little more than the elements of algebra are presupposed. The first four chapters treat those fundamental concepts and results which are essential in a more advanced study of any branch of ring theory. The rest of the monograph deals with somewhat more specialized results which, however, are of fairly wide interest and application. Natu- rally, many other topics of equal significance have had to be omitted simply because of space limitations. For the most part, material has been chosen which does not require any finiteness assumptions on the rings considered—the concept of a chain condition is not even mentioned until late in the last chapter! Chapter V consists primarily of an exposition of Krull's results on prime ideals in a commutative ring. Since no vii viii PREFACE finiteness conditions are imposed, it is necessary to use transfinite methods in establishing the existence of the prime ideals. Actually, Zorn's Maximum Principle is all that is required, and this is carefully stated and illustrated at an appropriate point in the chapter. An exposition of the theory of subdirect sums of rings is given in Chapter VI, together with a brief introduction to the Jacobson radical of a ring. Chapter VII is concerned with Boolean rings and with a number of algebraic generalizations of these rings. In particular, Stone's theorem on the representation of any Boolean ring as a ring of subsets of some set is obtained quite easily from results of the preceding chapter. The theory of matrices with elements in a commutative ring is the subject of Chapter VIII. Obviously, in one brief chapter it is impossible to give a comprehensive account of the theory. However, this short exposition has been included partly because of the generality of the approach, and partly because the material furnishes illustra- tions of a number of fundamental methods in the elementary theory of rings. This chapter is independent of the two preceding chapters. Finally, Chapter IX is an introduction to the study of primary ideals in a commutative ring. It ends with a short discussion of Noetherian rings (rings with ascending chain condition) and a brief section on algebraic manifolds. Since a considerable part of the material of this monograph has not previously appeared in book form, it has seemed desirable to give somewhat more complete references than might otherwise be the case. Accordingly, at the end of each chapter there appears a list of references to supplement the general material of the chapter, and also occasional source references to some of the more specific results which are not adequately treated in the PREFACE ix available treatises. The bibliography at the back of the book is by no means complete; for the most part, only those items have been included to which reference is made at some point in the text. It would be impossible to list here all the books and articles to which I am indebted. However, it may not be out of order to mention van der Waerden's Moderne Algebra which has played an important role in the develop- ment of widespread interest in abstract algebra. In so far as content is concerned, much of the first four chapters of this monograph, and a small part of the last, cover material which is included in van der Waerden's treatise. I am greatly indebted to Professors Bailey Brown, R. E. Johnson, and Saunders MacLane, who have read the entire manuscript and made many valuable suggestions. In particular, Professor Brown has taken an active interest in this project from its very inception and I have profited much by many discussions with him. Not only has he read the manuscript in all its versions in an unusually careful and critical manner, but he also has been of invaluable assistance in reading the proofs. NEAL HENRY MCCOY

Northampton, Mass. August 22, 1947

TABLE OF CONTENTS

CHAPTER PAGE I. Definitions and Fundamental Properties 1. Definition of a ring 1 2. Examples of rings 2 3. Properties of addition 10 4. Further fundamental properties 13 5. Division rings and fields 19 6. Equivalence relations 27 II. Polynomial Rings 7. Definitions and simple properties 31 8. Division transformation. Factor and remainder theorems 36 9. Polynomials with coefficients in a field 42 10. Rings of polynomials in several indeterminates . 46 III. Ideals and Homomorphisms 11. Ideals 52 12. Ideals generated by a finite number of elements . 54 13. Residue class rings 61 14. Homomorphisms and isomorphisms 67 15. Additional remarks on ideals 77 16. Conditions that a residue class ring be a field . .. 79 IV. Some Imbedding Theorems 17. A fundamental theorem 83 18. Rings without unit element 87 19. Rings of quotients 88 V. Prime Ideals in Commutative Rings 20. Prime ideals 96 21. The radical of an ideal 98 22. A maximum principle 100 23. Minimal prime ideals belonging to an ideal . . . 103 24. Maximal prime ideals belonging to an ideal . . . 107 VI. Direct and Subdirect Sums 25. Direct sum of two rings 114 26. Direct sum of any set of rings 118 27. Subrings of direct sums and subdirect sums . . . 120 28. A finite case 124 29. Subdirectly irreducible rings 128 30. The Jacobson radical and subdirect sums of fields 132

xi xii TABLE OF CONTENTS

CHAPTER PAGE VII. Boolean Rings and Some Generalizations 31. Algebra of logic and algebra of classes 138 32. Boolean rings 140 33. The p-rings 144 34. Regular rings 147 VIII. Rings of Matrices 35. Introduction 150 36. Definitions and fundamental properties 151 37. Determinants and systems of linear homogeneous equations 155 38. Characteristic ideal and null ideal of a matrix . 162 39. Resultants. 168 40. Divisors of zero 175 IX. Further Theory of Ideals in Commutative Rings 41. Primary ideals 180 42. The intersection of primary ideals 183 43. The prime ideals belonging to α 187 44. Short representation of an ideal 190 45. Noetherian rings 193 46. Ideals and algebraic manifolds 202

Bibliography 210 Index 215 BIBLIOGRAPHY Albert, A. A. [1] Modern Higher Algebra, Chicago, 1937. [2] Introduction to Algebraic Theories, Chicago, 1941. Artin, E., Nesbitt, C. J. and Thrall, R. M. [1] Rings with Minimum. Condition, Ann Arbor, 1944. Birkhoff, G. [1] Subdirect unions in universal algebra, Bull. Amer. Math. Soc, vol. 50 (1944), pp. 764-768. Birkhoff, G. and MacLane, S. [1] A Survey of Modern Algebra, New York, 1941. Bochner, S. [1] Lectures on Commutative Algebra, Princeton Univ. Lec- tures, 1937-38 (Planographed). Bourbaki, N. |1] Elements de malhemalique. Part I. Les structures fonda- mentales de I'analyse. Livre II. Algebre. Chapitre I. Structures algebriques (Actual Sci. Ind., no. 934), Paris, 1942. Brown, B. and McCoy, Ν. H. [1] Rings with unit element which contain a given ring. Duke Math. Jour., vol. 13 (1946), pp. 9-20. [2] Radicals and subdirect sums, Amer. Jour. Math., vol. 69 (1947), pp. 46-58. Busemann, H. [1] Introduction to Algebraic Manifolds, Princeton Univ. Press, 1939 (Planographed). Dickson, L. E. [1] Lin-ear Groups with an Exposition of the Galois Field Theory, Leipzig, 1901. [2] Modern Algebraic Theories, Chicago, 1926. Dubreil, P. [1] Algebre. Vol. 1. Equivalences, operations, groupes, anneaux, corps. (Cahiers scientifiques, no. 20), Paris, 1946. Forsythe, A. [1] Divisors of zero in polynomial rings, Amer. Math. Monthly, vol. 50 (1943), pp. 7-8. Forsythe, A. and McCoy, Ν. II. [1] On the commutativity of certain rings, Bull. Amer. Math. Soc, vol. 52 (1946), pp. 523-526. 210 BIBLIOGRAPHY 211

Fraenkel, A. [1] Uber die Teiler der Null und der Zerlegung von Ringen, Jour. Reine Angew. Math., vol. 145 (1915), pp. 139-176. Grell, H. [1] Beziehungen zwischen Idealen versckiedener Ringe, Math. Ann., vol. 97 (1927), pp. 490-523. Halmos, P. R. (1) Finite Dimensional Vector Spaces, Annals of Mathematics Studies, no. 7, Princeton, 1942. Jacobson. N. [1] The Theory of Rings, New York, 1943. [2] The radical and semi-simplicity for arbitrary rings, Amer. Jour. Math., vol. 67 (1945), pp. 300-320. [3] Structure theory for algebraic algebras of bounded degree, Annals of Math., vol. 46 (1945), pp. 695-707. Kothe, G. [1J Abstrdkte Theorie nichtkommulativer Ringe mil einer AnwendungaufdieDarstellungstkeoriekontinuierlicherGruppen, Math. Ann., vol. 103 (1930), pp. 545-572. Krull, W. [1] Ein neuer Beweis fur die Hauptsatze der allgemeinen Ideal- theorie, Math. Ann., vol. 90 (1923), pp. 55-64. [2] Idealtheorie in Ringen ohne Endlichkeitsbedingung, Math. Ann., vol. 101 (1929), pp. 729-744. [3] Idealtheorie, Berlin, 1935. [4] Enzyklopadie der Mathematischen Wissenschaften (second edition), vol. I, Part B, Algebra: I, 1, 11: Allgemeine Modul-, Ring- und Idealtheorie, I, 1, 12: Theorie der Polynomideale und Eliminations- theorie, Leipzig and Berlin, 1939. Macaulay, F. S. [1] The Algebraic Theory of Modular Systems, Cambridge Tracts in Mathematics arid Mathematical Physics, no. 19, London, 1916. MacDuffee, C. C. [1] The Theory of Matrices, Berlin, 1933. (2] An Introduction to Abstract Algebra, New York, 1940. [3] Vectors and Matrices, Carus Mathematical Monographs, no. 7, Ithaca, 1943. 212 BIBLIOGRAPHY

McCoy, Ν. Η. [1] Subrings of infinite direct sums, Duke Math. Jour., vol. 4 (1938), pp. 486-494. [2] Concerning matrices with elements in a commutative ring, Bull, Amer. Math. Soc, vol. 45 (1939), pp. 280-284. [3] Divisors of zero in matric rings, Bull. Amer. Math. Soc, vol. 47 (1941), pp. 166-172. [4] Remarks on divisors of zero, Amer. Math. Monthly, vol. 49 (1942), pp. 286-295. [5] Subdirectly irreducible commutative rings, Duke Math. Jour., vol. 12 (1945), pp. 381-387. [6] Subdirect sums of rings, Bull. Amer. Math. Soc, vol. 53 (1947), pp. 856-877. McCoy, Ν. H. and Montgomery, D. [1] A representation of generalized Boolean rings, Duke Math. Jour., vol. 3 (1937), pp. 455-159. von Neumann, J. [1] On regular rings, Proc. Nat. Acad. Sci., vol. 22 (1936), pp. 707-713. [2] Continuous Geometry, Princeton Univ. Lectures, 1936-37 (Planographed) Noether, E. [1] Idealtheorie in Ringbereichen, Math. Ann., vol. 83 (1921), pp. 24-66. Stone, Μ. H. [1] The theory of representations for Boolean algebras, Trans. Amer. Math. Soc, vol. 40 (1936), pp. 37-111.. [2] The Theory of Real Functions, Harvard Univ. Lectures, 1940 (Lithoprinted). Tukey, J. W. [1] Convergence and Uniformity in Topology, Annals of Mathe- matics Studies, no. 2, Princeton, 1940. van der Waerden, B. L. [1] Moderne Algebra, vols. 1 and 2, Berlin, 1930 and 1931; 2nd edition, vol. 1, 1937; vol. 2, 1940. [2] Algebraische Geometrie, Berlin, 1939. Reprinted, New York, 1945. Wedderburn, J. Η. M. [1] A theorem on finite algebras, Trans. Amer. Math. Soc, vol. 6 (1905), pp. 349-352. BIBLIOGRAPHY 213

[2] Lectures on Matrices, American Mathematical Society Colloquium Publications, vol. 17, New York, 1934. Weil, A. [1] Foundations of Algebraic Geometry, American Mathemati- cal Society Colloquium Publications, vol. 29, New York, 1946. Zariski, O. [1] A new proof of Hubert's Nullstellensatz, Bull. Amer. Math. Soc, vol. 53 (1947), pp. 362-368. Zorn, M. [1] A remark on method in Iransfinite algebra, Bull. Amer. Math. Soc, vol. 41 (1935), pp. 667-670.

INDEX OF TERMS

Page Page Abelian group 10 Galois field 23 Abstractly identical rings 72 Greatest common divisor . 42 Addition 1 Adjoint 157 Homomorphic image 68 Algebra of logic 138 Homomorphism 68 Algebraic manifold 203 Annihilator 159 Ideal 52 Ascending chain condition 193 generated by an element 55 Associative laws 1 generated by several elements 58 Basis of an ideal 58 of a manifold 203 Boolean algebra 138 Image of an element 68 ring 140 of an ideal 76 Indeterminate 31 Characteristic 2-1 Integral domain 15 ideal 163 Intersection of sets 5 polynomial 163 of ideals 77 Commutative elements... 9 Inverse 16 law 1 image 76 ring 9 Irreducible ideal 199 Complement 96 manifold 205 Complete direct sum 124 polynomial 44 Congruence modulo an Irredundant representa- ideal 62 tion 184 modulo an integer 28 Isolated component 192 Constant term 33, 48 primary component. . . . 193 Contain 54, 100 Isomorphic image 70 Isomorphism 70 Degree 33, 47, 48 Jacobson radical 132, 133 Direct sum 114, 118 Distributive laws 1 Division ring 19 Kernel 74 transformation 36 Divisor of zero 14 Leading coefficient 33 Left factor (divisor) 40 remainder 38 Equality 27 Linear system 101 Equivalence classes 29 relation 28 Manifold 203 of an ideal 203 Factor theorem 40 Matrix 151, 158 Field 21 Maximal element 100 Finite field 23 ideal in a ring...... 80, 100 Fundamental theorem of prime ideal belonging to algebra 86 an ideal 110 215 216 INDEX

Page Page Maximum principle 101 Residue class 64 Meet 104 class ring 64 Minimum function 163 Resultant 171 Minimal prime ideal be- Right factor (divisor).... 40 longing to an ideal. . 103 remainder 38 Monic polynomial 33 Ring 1 Multiplication 1 of finite subsets 7 Multiplicative system.... 89 of integers modulo n... 3 of matrices 8, 153 Nilpotent element 100 of quotients 88, 94 Noetherian ring 196 of real quaternions 8 Noncommutative ring.... 9 of subsets 4, 139 Null ideal 162 Root 40 One-to-one correspond- Scalar matrix 153 ence 70 Short representation 191 Skew field 19 Polynomial 31 Solution 202 Primary ideal 180 Subdirectly irreducible... 128 Prime ideal 96 reducible 128 ideal belonging to an Subdirect sum 121 ideal 182, 190 Subring 2, 52 Principal ideal 65 Subset 4 p-ring 144 Sum of sets 5, 139 Product of sets 5, 139 of elements 1 of elements 1 of ideals 78 Proper divisor of zero... 14 Symmetry, law of 28 Quasi-regular element. . . 132 Term 47 Radical ideal 201 Transitive law 28 of an ideal 99 Two-sided ideal 53 of a ring 124 Rank of a matrix 159 Union of ideals 77 Reducible ideal 199 of sets 100 manifold 205 Unit element 15 polynomial 44 Unrelated to an ideal. ... 108 Reflexive law 28 Regular ring 147 Void set 5 Related to an ideal.. 107, 108 Remainder theorem 39 Zero element 10, 11 Representation as subdi- ideal 53 rect sum 128 Zorn's Lemma 101 AMS / MAA THE CARUS MATHEMATICAL MONOGRAPHS

This monograph presents an introduction to that branch of abstract algebra having to do with the theory of rings, with some emphasis on the role of ideals in the theory.

Except for a knowledge of certain fundamental theorems about determinants which is assumed in Chapter VIII, and at one point in Chapter VII, the book is almost entirely self- contained. Of course, the reader must have a certain amount of “mathematical maturity” in order to understand the illustrative examples, and also to grasp the signifi cance of the abstract approach. However, in so far as formal technique is concerned, little more than the elements of algebra are presupposed.