16
Problem Books in Mathematics Edited by Peter Winkler
Problem Books in Mathematics
Edited by Peter Winkler
Problem Books in Mathematics Series Editors: Peter Winkler
Pell's Equation by Edward J. Barbeau
Polynomials by Edward J. Barbeau
Problems in Geometry by Marcel Berger, Pierre Pansu, Jean-Pic Berry, and Xavier Saint-Raymond
Problem Book for First Year Calculus by George W. Bluman
Exercises in Probability by T. Cacoullos
Probability Through Problems by Marek Capinski and Tomasz Zastawniak
An Introduction to Hilbert Space and Quantum Logic by David W. Cohen
Unsolved Problems in Geometry by Hallard T. Croft, Kenneth J. Falconer, and Richard K. Guy
Berkeley Problems in Mathematics (Third Edition) by Paulo Ney de Souza and Jorge-Nuno Silva
The IMO Compendium: A Collection of Problems Suggested for the International Mathematical Olympiads: 1959-2004 by Dusan Djukic, Vladimir Z. Jankovic, Ivan Matic, and Nikola Petrovic
Problem-Solving Strategies by Arthur Engel
Problems in Analysis by Bernard R. Gelbaum
Problems in Real and Complex Analysis by Bernard R. Gelbaum
(continued after subject index)
T.Y. Lam
Exercises in Modules and Rings
~ Springer
T.Y. Lam Department of Mathematics University of California, Berkeley Berkeley, CA 94720-0001 USA [email protected]
Series Editor: Peter Winkler Department of Mathematics Dartmouth College Hanover, NH 03755 USA Peter. [email protected]
Mathematics Subject Classification (2000): 00A07 13-XX 16-01
Library of Congress Control Number: 2006935420
ISBN-lO: 0-387-98850-5 ISBN-13: 978-0-387-98850-4
Printed on acid-free paper.
e-ISBN-10: 0-387-48899-5 e-ISBN-13: 978-0-387-48899-8
© 2007 Springer Science+ Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part
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To Chee King
A constant source of strength and inspiration
Preface
The idea of writing this book came roughly at the time of publication of my graduate text Lectures on Modules and Rings, Springer GTM Vol. 189, 1999. Since that time, teaching obligations and intermittent intervention of other projects caused prolonged delays in the work on this volume. Only a lucky break in my schedule in 2006 enabled me to put the finishing touches on the completion of this long overdue book.
This book is intended to serve a dual purpose. First, it is designed as a "problem book" for Lectures. As such, it contains the statements and full solutions of the many exercises that appeared in Lectures. Second, this book is also offered as a reference and repository for general information in the theory of modules and rings that may be hard to find in the standard textbooks in the field.
As a companion volume to Lectures, this work covers the same mathematical material as its parent work; namely, the part of ring theory that makes substantial use of the notion of modules. The two books thus share the same table of contents, with the first half treating projective, injective, and flat modules, homological and uniform dimensions, and the second half dealing with noncommutative localizations and Goldie's theorems, maximal rings of quotients, Frobenius and quasi-Frobenius rings, concluding with Morita's theory of category equivalences and dualities. Together with the coverage of my earlier text First Course in Noncommutative Rings, Springer GTM Vol. 131, these topics comprise a large part of the foundational material in the classical theory of rings.
An integral part of Lectures is the large collection of six hundred exercises distributed over the seven chapters of the book. With the exception of two or three (which are now deemed too difficult for inclusion), all exercises are solved in full in this problem book. Moreover, some 40 new exercises have been added to the present collection to further broaden its
viii Preface
coverage. To facilitate the cross-referencing, I have by and large used the same numbering scheme for the exercises in the two books. Some exceptions to this rule are explained in the Notes to the Reader section on page xiii.
Problem solving is something truly special in mathematics. Every student trying to learn a mathematical subject with any degree of seriousness finds it helpful or even necessary to do a suitable number of exercises along the way, to help consolidate his or her understanding of the subject matter, and to internalize the myriad of information being offered. This exercise book is intended not to supplant this process, but rather, to facilitate it. There are certainly more exercises in Lectures than the author can realistically expect his readers to do; for instance, §3 alone contains as many as 61 exercises. If my teaching experience is any guide, most students appreciate doing some exercises in detail, and learning about others by reading. And, even in cases where they solved exercises on their own, they find it helpful to compare their solutions with more "official" versions of the solutions, say prepared by the author. This is largely the raison d' etre of a problem book, such as the present one.
What this book offers, however, is more than exercise solutions. Among the exercises in Lectures, only a rather small number are of a routine nature. The others range from nontrivial to medium-difficult, difficult, challenging, to very challenging, although they are not explicitly identified as such. In quite a few cases, the "exercises" are based on original results of other authors published in the research literature, for which no convenient expositions are available. This being the situation, a problem book like this one where all exercise solutions are independently written and collected in one place should be of value to students and researchers alike. For some problems that can be approached from several different angles, sometimes more than one solution is given. Many of the problem solutions are accompanied by a Comment section giving relevant bibliographical, historical or anecdotal information, pointing out latent connections to other exercises, or offering ideas on further improvements and generalizations. These Comment sections rounding out the solutions of the exercises are intended to be a particularly useful feature of this problem book.
This book is an outgrowth of my lecture courses and seminars over the years at the University of California at Berkeley, where many of the problem solutions were presented and worked over. As a result, many of my students and seminar participants have offered corrections and contributed useful ideas to this work; I thank them all. As usual, the warm support of my family (Chee King; Juwen, Fumei, Juleen, and Tsai Yu) was instrumental to the completion of this project, for which no words of acknowledgement could be adequate enough.
Berkeley, California September 25, 2006
T.Y.L.
Contents
Preface ........ . Vll
Notes to the Reader . . . xi
Partial List of Notations .... Xlll
Partial List of Abbreviations .............. . XVll
1 Free Modules, Projective, and Injective Modules . . . . . 1
§l. Free Modules . . . . . . . . . . . . . . . . . . . . . . . . . 1
§2.
§3.
37 Exercises Projective Modules 46 Exercises Injective Modules . 61 Exercises
21
60
2 Flat Modules and Homological Dimensions . . . . . . . .. 97
§4. Flat Modules . . . . . . . . . . . . . . . . . . . . . . . .. 97
§5. 57 Exercises Homological Dimensions 36 Exercises
3 More Theory of Modules
§6. Uniform Dimensions, Complements, and CS Modules. 45 Exercises
131
155 155
x Contents
§7. Singular Submodules and Nonsingular Rings ....... 185 41 Exercises
§8. Dense Submodules and Rational Hulls . . . . . . . . . .. 205 24 Exercises
4 Rings of Quotients .
§9. Noncommutative Localization 6 Exercises
217
217
§1O. Classical Ring of Quotients . . . . . . . . . . . . . . . .. 222 37 Exercises
§11. Right Goldie Rings and Goldie's Theorem . . . . . . . .. 242 33 Exercises
§12. Artinian Rings of Quotients . . . . . . . . . . . . . . . .. 260 12 Exercises
5 More Rings of Quotients
§13. Maximal Rings of Quotients. 31 Exercises
§14. Martindal's Ring of Quotients. 17 Exercises
6 Frobenius and Quasi-Frobenius Rings
§15. Quasi-Frobenius Rings ....... . 25 Exercises
§16. Frobenius Rings and Symmetric Algebras 35 Exercises
271
271
287
299 299
315
7 Matrix Rings, Categories of Modules and Morita Theory 343
§17. Matrix Rings . . . . . . . . . . . . . . . . . . . . . . . .. 343 16 Exercises
§18. Morita Theory of Category Equivalences. . . . . . . . .. 353 39 Exercises
§19. Morita Duality Theory. . . . . . . . . . . . . . . . . . .. 378 41 Exercises
Name Index .. 403
Subject Index . 407
Notes to the Reader
Since this Problem Book is based on the author's "Lectures on Modules and Rings", the two books share a common organization. Thus, just as in Lectures, the main text of this book contains seven chapters, which are divided into nineteen sections. For ease of reference, the sections are numbered consecutively, independently of the chapters, from §1 to §19. The running heads offer the quickest and most convenient way to tell what chapter and what section a particular page belongs to. This should make it very easy to find an exercise with a given number.
Each section begins with its own introduction, in which the material in the corresponding section in Lectures is briefly recalled and summarized. Such short introductions thus serve as recapitulations of the theoretical underpinnings of the exercises that follow. The exercises in §8, for instance, are numbered 8.1, 8.2, etc., with occasional aberrations such as 8.5A and 8.5B. The exercise numbers are almost always identical to those in Lectures, though in a few cases, some numbers may have shifted by one. Exercises with numbers such as 8.5A, 8.5B are usually added exercises that did not appear before in Lectures.
For the exercise solutions, the material in Lectures is used rather freely throughout. A code such as LMR-(3.7) refers to the result (3.7) in Lectures on Modules and Rings. Occasional references are also made to my earlier Springer books FC (First Course in Noncommutative Rings, 2nd ed., 2001) and ECRT (Exercises in Classical Ring Theory, 2nd ed., 2003). These are usually less essential references, included only for the sake of making further connections.
The ring theory conventions used in this book are the same as those introduced in LMR. Thus, a ring R means a ring with identity (unless
xii Notes to the Reader
otherwise stated). A subring of R means a subring containing the identity of R (unless otherwise stated). The word "ideal" always means a two-sided ideal; an adjective such as "noetherian" likewise means both right and left noetherian. A ring homomorphism from R to S is supposed to take the identity of R to that of S. Left and right R-modules are always assumed to be unital; homomorphisms between modules are (usually) written on the opposite side of the scalars. "Semisimple rings" are in the sense of Wedderburn, Noether and Artin: these are rings R that are semisimple as (left or right) modules over themselves. Rings with Jacobson radical zero are called Jacobson semisimple (or semiprimitive) rings.
Throughout the text, we use the standard notations of modern mathematics. For the reader's convenience, a partial list of the ring-theoretic notations used in this book is given on the following pages.
Partial List of Notations
Z N Q lR C Zn Gpoo o c, ~
<;; IAI, CardA A\B A >---+ B A -7t B Dij
Eij
M t , MT Mn(S) GLn(S) GL(V) Z(G) GG(A) [G:H] [K:F] mR, Rm mig Igm
R ' R
the ring of integers the set of natural numbers the field of rational numbers the field of real numbers the field of complex numbers the ring (or the cyclic group) ZjnZ the Priifer p-group the empty set used interchangeably for inclusion strict inclusion used interchangeably for cardinality of A set-theoretic difference injective mapping from A into B surjective mapping from A onto B Kronecker deltas standard matrix units transpose of the matrix M set of n x n matrices with entries from S group of invertible n x n matrices over S group of linear automorphisms of a vector space V center of the group (or the ring) G centralizer of A in G index of subgroup H in a group G field extension degree category of right (left) R-modules category of f.g. right (left) R-modules
xiv Partial List of Notations
MR,RN RMS
M0RN HomR(M,N) EndR(M) nM M(I)
MI An(M) soc(M) rad(M) Ass(M) E(M) E(M) Z(M) length M u.dim M rank M p(M), PR(M) rkM M* M', MO M M' , N**, cl(N) N<;;:e M N<;;:d M N~cM ROP
U(R), R* U(D), D*, iJ CR
C(N) radR Nil*R Nil*R Nil (R) A£(R), AT(R) Max(R) Spec(R) I(R)
soc( RR) , soc(RR) Z(RR), Z(RR) Pic(R)
right R-module M, left R-module N (R, S)-bimodule M tensor product of MR and RN group of R-homomorphisms from M to N ring of R-endomorphisms of M M ffi ... ffi M (n times) EBiEI M (direct sum of I copies of M) DiEI M (direct product of I copies of M) n -th exterior power of M socle of M radical of M set of associated primes of M injective hull (or envelope) of M rational hull (or completion) of M singular submodule of M (composition) length of M uniform dimension of M torsionfree rank or (Goldie) reduced rank of M p-rank of MR
rank (function) of a projective module M R-dual of an R-module M character module Homz(M, QjZ) of MR k-dual of a k-vector space (or k-algebra) M Goldie closure of a submodule N ~ M N is an essential submodule of M N is a dense sub module of M N is a closed submodule of M the opposite ring of R group of units of the ring R multiplicative group of the division ring D set of regular elements of a ring R set of elements which are regular modulo the ideal N Jacobson radical of R upper nilradical of R lower nilradical (a.k.a. prime radical) of R nilradical of a commutative ring R left, right artinian radical of R set of maximal ideals of a ring R set of prime ideals of a ring R set of isomorphism classes of indecomposable injective modules over R right (left) socle of R right (left) singular ideal of R Picard group of a commutative ring R
Rs RS-l, S-IR Rp Q:-nax(R) , Q~ax(R) Q~I(R), Q~I(R) Qcl(R), Q(R) Qr(R), QI(R) QS(R) annr(S), annl(S) annM(S) kG, k[G]
k[Xi : iEI]
k(Xi : iEI) k[[Xl, . .. ,xnll
Partial List of Notations xv
universal S-inverting ring for R right (left) Ore localization of R at S localization of (commutative) R at prime ideal p maximal right (left) ring of quotients for R classical right (left) ring of quotients for R the above when R is commutative Martindale right (left) ring of quotients symmetric Martindale ring of quotients right, left annihilators of the set S annihilator of S taken in M (semi)group ring of the (semi)group G over the ring k polynomial ring over k with commuting variables {Xi : i E I} free ring over k generated by {Xi : i E I} power series ring in the Xi's over k
Partial List of Abbreviations
RHS, LHS ACC DCC IBN PRIR, PRID PLIR, PLID FFR QF PF PP PI CS QI Obj iff resp. ker coker im rk f.cog. f.g. f.p. f.r. l.c.
right-hand side, left-hand side ascending chain condition descending chain condition "Invariant Basis Number" property principal right ideal ring (domain) principal left ideal ring (domain) finite free resolution quasi-Frobenius pseudo-Frobenius "principal implies projective" "polynomial identity" (ring, algebra) "closed submodules are summands" quasi-injective (module) objects (of a category) if and only if respectively kernel cokernel image rank finitely cogenerated finitely generated finitely presented finitely related linearly compact
XVlll Partial List of Abbreviations
pd id fd wd r.gl.dim l.gl.dim
projective dimension injective dimension flat dimension weak dimension (of a ring) right global dimension (of a ring) left global dimension (of a ring)
