i Lommutdtive Revised Edition Rings IRVING KAPLANSKY ---_I- The University of Chicago Press Chicago and London Lontents To Chellie Preface ix Preface to the Revised Edition xi IRVING KAPLANSKY is George Herbert Mead Chapter 1 Prime Ideals and Integral Extensions 1 Distinguished Service Professor in the 1-1 Prime Ideals 1 Department of Mathematics, the University 1-2 Integral Elements, I 8 of Chicago. He is the author of numerous 1-3 G-ideals, Hilbert Rings, and the Nullstellensatz 12 books and articles and is the general editor 1-4 Localization 22 of the Chicago Lectures in Mathematics series published by the University of Chicago Press. 1-5 Prime Ideals in Polynomial Rings 25 [1974] 1-6 Integral Elements, I1 27 Chapter 2 Noetherian Rings 47 2-1 The Ascending Chain Condition 47 2-2 Zero-divisors 54 2-3 Integral Elements, I11 66 The University of Chicago Press, Chicago 60637 The University of Chicago Press, Ltd., London 2-4 Intersections of Quasi-local Domains 75 @ 1970, 1974 by The University of Chicago Chapter 3 Macaulay Rings and Regular Rings 84 All rights reserved. Revised edition published I974 3-1 R-sequences and Macaulay Rings 84 Printed in the United States of America 3-2 The Principal Ideal Theorem 104 International Standard Book Number: 0-226-42454-3 3-3 Regular Rings 115 Library of Congress Catalog Card Number: 71-102759 Chapter 4 Homological Aspects of Ring Theory 123 4-1 Homology 123 4-2 Unique Factorization 130 4-3 The Euler Characteristic 137 4-4 Change of Rings for Injective Dimension 149 4-5 Gorenstein Rings 156 4-6 Duality 165 Notes 169 Bibliography 171 Index of Theorems 174 Index of Topics 181 V PreFdce This account of commutative rings has grown over the years through various stages. The first version was an appendix to the notes on homo- logical dimension issued in 1959 (these notes, without the appendix, have now appeared as Part I11 of [26]). The second was a crude 49 page dittoed manuscript written in 1961. This account was expanded in the 1965-6 course presented at Queen Mary College. I owe an enormous debt to Professor Paul M. Cohn of Bedford College for his expert job in writing up the course; large parts are incorporated here with only small changes. A draft of the book was prepared during the summer of 1968, and was used for a course at Chicago during 1968-9. No attempt has been made to achieve scholarly completeness. Refer- ences to sources have been made only in scattered instances where it seemed particularly desirable, and the bibliography contains only items to which there is an actual reference. My intention is to give an account of some topics in the theory of commutative rings in a way that is acces- sible to a reader with a modest background in modern algebra; I assume only an acquaintance with the definitions and most elementary proper- ties of rings, ideals, and modules. More exactly, this is true till $4-1, where I presuppose the theory of homological dimension as developed in [26], and 54-4 where use of the long exact sequence for Ext begins. I hope that readers will find it feasible to go on from this book to a deeper study of the literature. It would be most urgent to learn the theory of completion and the Cohen structure theorems. I have inserted numerous exercises, partly to cover additional ma- terial of lesser importance, and partly to give the reader a chance to test his growing skill. There is occasional reliance on the exercises as part of the exposition, and I hope no r ader will find this inconvenient. In the style of Landau, or Hard ,and Wright, I have presented the s vii ... Vlll PREFACE material as an Unbroken series of theorems. I prefer this to the n-place decimal system favored by some authors, and I have also grown tired of seeing a barrage of lemmas, propositions, corollaries, and scholia (whatever they are). I admit that this way the lowliest lemma gets ele- vated to the same eminence as the most awesome theorem. Also, the number of theorems becomes impressive, so impressive that I felt the need to add an index of theorems. All rings have a unit element, except for a fleeting instant in Ex. 22 Preface of $2-2. All rings are commutative except in several (not quite so fleet- ing) isolated passages. Somewhat erratically, the adjective ~~~~~~~~~- tive” is occasionally inserted, merely for emphasis. to the Revised Edition I am very grateful to Joel Cohen, David Eisenbud, Graham Evans, Marshall Fraser, Ross Hamsher, William Heinzer, Martin Isaacs, Stanley Kochman, Peter Kohn, Stephen McAdam, Judith Sally, and Wolmer Vasconcelos for numerous valuable suggestions. If fully credited, their names would be everywhere dense. Robert Gilmer was In this reprinting of Commutative Rings minor slips have been corrected especially generous in reading the manuscript critically and enlightening and several more substantial changes have been made (largely in the me on many points. exercises). I am indebted to many students and colleagues; they are too Final thanks go to Joyce Bolden and Diane Moore for a splendid job numerous to list, but I am none the less appreciative. of typing, and to Sylvia Clark, Gerald Curtis, and Mary Johnson of In a brief new section entitled “Notes” I have added some comments Allyn and Bacon, Inc. for their fine cooperation. which are in the nature of afterthoughts rather than a revision of the text. IRVING UPLANSKY CHAPTER 1 Prime Ideals md Integral Extensions 1-1’ PRIME IDEALS I- >r Prime ideals pl$ a central role in the theory of commutative rings, and it is appropriate to devote the first section to a collection of ob- servations concerning them. We recall the definition: in a commutative ring R, the ideal I is prime if ab e I implies a e I or b E I. Alternatively: I is prime if R/I is an integral domain. The convention that R itself is not regarded as a prime ideal is fairly universally accepted now, and we shall adhere to it (except, no doubt, for occasional sllps). In the ring of integers there is an analogous convention that 1 is not a prime. Let S be the set-theoretic complement of an ideal I. Then the defini- tion of a prime ideal can be recast as follows: I is prime if and only if S is multiplicatively closed. Now it goes without saying that I is maximal with respect to the exclusion of .S,Krull discovered a very useful con- verse. “1 I Theorem 1. Let S be a multiplicatively closed set in a ring R and let I be an ideal in R maximal with respect to the exclusion of S. Then I is prime. Proof. Given ab e I we must show that a or b lies in I. Suppose the contrary. Then the ideal (I, a) generated by I and a is strictly larger 1 2 CH. I/PRIMEIDEALS AND INTEGRAL EXTENSIONS SEC. PRIME IDEALS 3 than I and therefore intersects S. Thus there exists an element s1 e S of 1. The set consisting of 1 is multiplicatively closed. Its saturation is the form s1 = il + xa (il e I, x e R). Similarly we have s2 e S, s2 = iz + yb. the set of all units. The prime ideals maximal in the complement are But then just the ordinary maximal ideals. 2. Let S be the set of all non-zero-divisors in R. S is a saturated (1) s1s2 = (il + xa)(i2 + yb) multiplicatively closed set. So: the- zerg-divisors in R are a union of prime ideals. The prime ideals‘maximal among these might be called and all four terms on the right of (1) lie in Z (the first three because a ‘the “maximal primes of zero-divisors.” ~ factor lies in Z and the fourth because ab e Z). Hence s1s2e Z, a contra- --j.”;:. diction. Both of these illustrations admit appropriate generalizations to C’ modules: in the first case take all elements of R that act one-to-one and We note that, given any ideal J disjoint from a multiplicatively closed onto as multiplications of the given R-module; in the second case take set S, we can by Zorn’s lemma expand J to an ideal Z maximal with all non-zero-divisors on the module. This second case is important respect to disjointness from S. Thus we have a method of constructing enough to merit a definition. prime ideals. Before proceeding, let us examine more closely the complement of Definition. Let R be any commutative ring, A any non-zero R- a prime ideal. It is not any old multiplicatively closed set; it has the module. The prime ideals maximal within the zero-divisors on A are further property of being saturated in the sense that along with an called the maximal primes of A. When A has the form R/I, I an ideal element x it containsLan the divisors of x. We are led to the following in R, we say maximal primes of I, rather than of R/Z; the danger of tLeorem, in essence a sharpenin6 of Theorem 1. ambiguity is slight. I %’ b #!, ;* n At this point we shall briefly discuss unique factorization domains Theorem 2. The following statements on a set S in a ring R are (UFD’s). They can be characterized by a suitable interplay between equivalent: (1) S is a saturated multiplicatively closed set, (2)the comple- prime ideals and principal ideals (Theorem 5).