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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 By NEAL H. McCOY Professor of Mathematics Smith College Published by THE MATHEMATICAL ASSOCIATION OF AMERICA Copyright ©1948 by The Mathematical Association of America Library of Congress Catalog Card Number: 48023679 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 under- stand the illustrative examples, and also to grasp the sig- nificance of the abstract approach. However, in so far as formal technique is concerned, little more than the ele- ments 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 generali- zations 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 pre- ceding 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 ap- proach, and partly because the material furnishes illustra- tions of a number of fundamental methods in the elemen- tary theory of rings. This chapter is independent of the two preceding chapters. Finally, Chapter IX is an introduc- tion 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 mono- graph has not previously appeared in book form, it has seemed desirable to give somewhat more complete refer- ences 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 .
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