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Xerox University Microfilms 300 North Zeeb Road Ann Arbor, Michigan 48106 76-18,862 INFORMATION TO USERS This material was produced from a microfilm copy of the original document. While the most advanced technological means to photograph and reproduce this document have been used, the quality is heavily dependent upon the quality of the original submitted. The following explanation of techniques is provided to help you understand markings or patterns which may appear on this reproduction. 1.The sign or "target" for pages apparently lacking from the document photographed is "Missing Page(s)". If It was possible to obtain the missing page(s) or section, they are spliced into the film along with adjacent pages. This may have necessitated cutting thru an image and duplicating adjacent pages to insure you complete continuity. 2. 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Xerox University Microfilms 300 North Zeeb Road Ann Arbor, Michigan 48106 76-18,862 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.
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