Rings and Categories of Modules

385 Pages·1992·9.689 MB·English

by  Frank W. Anderson, Kent R. Fuller (auth.)

13
Graduate Texts in Mathematics
Editorial Board
J.H. Ewing F.W. Gehring P.R. Halmos
Graduate Texts in Mathematics
1 TAKEUTI/ZARING. Introduction to Axiomatic Set Theory. 2nd ed.
2 OXTOBY. Measure and Category. 2nd ed.
3 SCHAEFFER. Topological Vector Spaces.
4 HILTON/STAMMBACH. A Course in Homological Algebra.
5 MAC LANE. Categories for the Working Mathematician.
6 HUGHES/PIPER. Projective Planes.
7 SERRE. A Course in Arithmetic.
8 TAKEUTI/ZARING. Axiometic Set Theory.
9 HUMPHREYS. Introduction to Lie Algebras and Representation Theory.
10 COHEN. A Course in Simple Homotopy Theory.
11 CONWAY. Functions of One Complex Variable. 2nd ed.
12 BEALS. Advanced Mathematical Analysis.
13 ANDERSON/FULLER. Rings and Categories of Modules. 2nd ed.
14 GOLUBITSKY/GUILEMIN. Stable Mappings and Their Singularities.
15 BERBERIAN. Lectures in Functional Analysis and Operator Theory.
16 WINTER. The Structure of Fields.
17 ROSENBLATT. Random Processes. 2nd ed.
18 HALMOS. Measure Theory.
19 HALMOS. A Hilbert Space Problem Book. 2nd ed., revised.
20 HUSEMOLLER. Fibre Bundles. 2nd ed.
21 HUMPHREYS. Linear Algebraic Groups.
22 BARNES/MACK. An Algebraic Introduction to Mathematical Logic.
23 GREUB. Linear Algebra. 4th ed.
24 HOLMES. Geometric Functional Analysis and Its Applications.
25 HEWITT/STROMBERG. Real and Abstract Analysis.
26 MANES. Algebraic Theories.
27 KELLEY. General Topology.
28 ZARISKI/SAMUEL. Commutative Algebra. Vol. I.
29 ZARISKI/SAMUEL. Commutative Algebra. Vol. II.
30 JACOBSON. Lectures in Abstract Algebra I. Basic Concepts.
31 JACOBSON. Lectures in Abstract Algebra II. Linear Algebra.
32 JACOBSON. Lectures in Abstract Algebra III. Theory of Fields and Galois Theory.
33 HIRSCH. Differential Topology.
34 SPITZER. Principles of Random Walk. 2nd ed.
35 WERMER. Banach Algebras and Several Complex Variables. 2nd ed.
36 KELLEY!NAMIOKA et al. Linear Topological Spaces.
37 MONK. Mathematical Logic.
38 GRAUERT/FRITZSCHE. Several Complex Variables.
39 ARVESON. An Invitation to C' -Algebras.
40 KEMENy/SNELL/KNAPP. Denumerable Markov Chains. 2nd ed.
41 ApOSTOL. Modular Functions and Dirichlet Series in Number Theory. 2nd ed.
42 SERRE. Linear Representations of Finite Groups.
43 GILLMAN/JERISON. Rings of Continuous Functions.
44 KENDIG. Elementary Algebraic Geometry.
45 LOEVE. Probability Theory I. 4th ed.
46 LOEVE. Probability Theory II. 4th ed.
47 MOISE. Geometric Topology in Dimentions 2 and 3.
cUIJtinued after index
Frank W. Anderson Kent R. Fuller
Rings and
Categories of Modules
Second Edition
Springer-Verlag
New York Berlin Heidelberg London Paris
Tokyo Hong Kong Barcelona Budapest
Frank W. Anderson Kent R. Fuller
University of Oregon University of Iowa
Department of Mathematics Department of Mathematics
Eugene, OR 97403 Iowa City, IA 52242
USA USA
Editorial Board
J.H. Ewing F.W. Gehring P.R. Halmos
Department of Department of Department of
Mathematics Mathematics Mathematics
Indiana University University of Michigan Santa Clara University
Bloomington, IN 47405 Ann Arbor, MI 48109 Santa Clara, CA 95053
USA USA USA
Mathematics Subject Classifications (1991): 13-01, 16-01
Library of Congress Cataloging-in-Publication Data
Anderson, Frank W. (Frank Wylie), 1928-
Rings and categories of modules / Frank W. Anderson. Kent R.
Fuller.-2nd ed.
p. cm.-(Graduate texts in mathematics; 13)
Includes bibliographical references and index.
ISBN-13: 978-1-4612-8763-6 e-ISBN-13: 978-1-4612-4418-9
DOl: 10.1007/978-1-4612-4418-9
I. Modules (Algebra) 2. Rings (Algebra) 3. Categories
(Mathematics) I. Fuller, Kent R. II. Title. III. Series.
QA247.A55 1992
512'.4-dc20 92-10019
Printed on acid-free paper.
~( 1974, 1992 Springer-Verlag New York, Inc.
Softcover reprint ofthe hardcover 2nd edition 1992
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9 8 7 6 543 2 I
Preface
This book is intended to provide a reasonably self-contained account of a
major portion of the general theory of rings and modules suitable as a text
for introductory and more advanced graduate courses. We assume the famil
iarity with rings usually acquired in standard undergraduate algebra courses.
Our general approach is categorical rather than arithmetical. The continuing
theme of the text is the study of the relationship between the one-sided
ideal structure that a ring may possess and the behavior of its categories of
modules.
Following a brief outline of set-theoretic and categorical foundations, the
text begins with the basic definitions and properties of rings, modules and
homomorphisms and ranges through comprehensive treatments of direct
sums, finiteness conditions, the Wedderburn-Artin Theorem, the Jacobson
radical, the hom and tensor functions, Morita equivalence and duality, de
composition theory of injective and projective modules, and semi perfect and
perfect rings. In this second edition we have included a chapter containing
many of the classical results on artinian rings that have hdped to form the
foundation for much of the contemporary research on the representation
theory of artinian rings and finite dimensional algebras. Both to illustrate the
text and to extend it we have included a substantial number of exercises
covering a wide spectrum of difficulty. There are, of course" many important
areas of ring and module theory that the text does not touch upon. For
example, we have made no attempt to cover such subjects as homology, rings
of quotients, or commutative ring theory.
This book has evolved from our lectures and research over the past
several years. We are deeply indebted to many of our students and colleagues
for their ideas and encouragement during its preparation. We extend our
sincere thanks to them and to the several people who have helped with the
preparation of the manuscripts for the first two editions, and/or pointed out
errors in the first.
Finally, we apologize to the many authors whose works we have used but
not specifically cited. Virtually all of the results in this book have appeared in
some form elsewhere in the literature, and they can be found either in the
books and articles that are listed in our bibliography, or in those listed in the
collective bibliographies of our citations.
Eugene, OR Frank W. Anderson
Iowa City, IA Kent R. Fuller
January 1992
v
Contents
Preface ....... .................... ...... .... ...... .. .... .. v
§o. Preliminaries ... ........ ..... .. ..... .... ... ..... .
Chapter 1: Rings, Modules and Homomorphisms .. .. .... ....... . 10
§1. Review of Rings and their Homomorphisms ... .... .. . 10
§2. Modules and Submodules ....... ...... ...... ...... . 26
§3. Homomorphisms of Modules ... .. ......... ... ..... . 42
§4. Categories of Modules; Endomorphism Rings .... ... .. 55
Chapter 2: Direct Sums and Products ... .... ..... .. ........ ... 65
§5. Direct Summands .... ... ............ ....... ...... 65
§6. Direct Sums and Products of Modules ... ............ 78
§7. Decomposition of Rings ........ ............ ....... 95
§8. Generating and Cogenerating .......... .. ...... ... . 105
Chapter 3: Finiteness Conditions for Modules .. ... ... ... .... ... 115
§9. Semisimple Modules- The Socle and the Radical .... . 115
§1O. Finitely Generated and Finitely Cogenerated Modules-
Chain Conditions .. .. ..... ...... .. .... ... ......... 123
§11. Modules with Composition Series .................. . 133
§12. Indecomposable Decompositions of Modules ... .... .. 140
Chapter 4: Classical Ring-Structure Theorems ........ .. .... .... 150
§13. Semisimple Rings ................................. 150
§14. The Density Theorem .................. ...... .. .. . 157
§15. The Radical of a Ring- Local Rings and Artinian Rings 165
Chapter 5: Functors Between Module Categories .. .. ... .... .. .. 177
§16. The Hom Functors and Exactness- Projectivity
and Injectivity .. .... .. ... ... ... .... .. .. .... .... .. 178
§17. Projective Modules and Generators ........... ..... . 191
vii
V1l1 Contents
§18. Injective Modules and Cogenerators .... ...... ..... .. 204
§19. The Tensor Functors and Flat Modules ... ...... ..... 218
§20. Natural Transformations .......... .... ......... .. . 234
Chapter 6: Equivalence and Duality for Module Categories ... .... 250
§21. Equivalent Rings ... ..... ...... ... .... .... .... .... 250
§22. The Morita Characterizations of Equivalence .. .. .. ... 262
§23. Dualities ... ........ ..... ........ .... ......... ... 269
§24. Morita Dualities ... .. .... ...... .. ............ .... 278
Chapter 7: Injective Modules, Projective Modules, and
Their Decompositions ... . . . . . . . . . . . . . . . . . . . . . . 288 . . . .
§25. Injective Modules and Noetherian Rings-
The Faith- Walker Theorems. . . . . . . . . . ... . .. . . . . . . 2. 88
§26. Direct Sums of Countably Generated Modules-
With Local Endomorphism Rings ... ... ..... ....... . 295
§27. Semi perfect Rings ............... ............ ..... 301
§28. Perfect Rings ..................... ........... .... 312
§29. Modules with Perfect Endomorphism Rings ...... ... . 322
Chapter 8: Classical Artinian Rings .......... ..... ........ .... 327
§30. Artinian Rings with Duality ........................ 327
§31. Injective Projective Modules ...... ........... ...... 336
§32. Serial Rings .... ...... .... ... ...... .... .... ...... . 345
Bibliography ...... ..... ........ .......... ............ ... ... 363
Index ...................................... .. ............ . 369
Rings and Categories of Modules
Preliminaries
§o.
Preliminaries
In this section is assembled a summary of various bits of notation, termin
ology, and background information. Of course, we reserve the right to use
variations in our notation and terminology that we believe to be self
explanatory without the need of any further comment.
A word about categories. We shall deal only with very special concrete
categories and our use of categorical algebra will be really just terminological
-at a very elementary level. Here we provide the basic terminology that we
shall use and a bit more. We emphasize though that our actual use of it will
develop gradually and, we hope, naturally. There is, therefore, no need to try
to master it at the beginning.
0.1. Functions. Usually, but not always, we will write functions "on the
left". That is, if J is a function from A to B, and if a E A, we write J(a) for the
value of J at a. Notation like J: A --+ B denotes a function from A to B. The
elementwise action of a functionJ: A --+ B is described by
J:a f--+ f(a) (a E A).
I
Thus, if A' <;: A, the restriction (f A') of J to A' is defined by
(fl A'): a' f--+ J(a') (a' E A').
GivenJ:A --+ B, A' <;: A, and B' <;: B, we write
I
J(A') = {f(a) a E A'} and
For the composite or product of two functionsJ: A --+ Band g: B --+ C we write
gof , or when no ambiguity is threatened, just gJ; thus, g 0 J: A --+ C is defined
by goJ : a f--+ g(f(a» for all a E A. The resulting operation on functions is
associative wherever it is defined. The identity Junction from A to itself is
denoted by lA' The set of all functions from A to B is denoted by BA or by
Map(A, B):
BA = Map(A, B) = {f I J: A --+ B}.
So AA is a monoid (= semigroup with identity) under the operation of
composition.
A diagram of sets and functions commutes or is commutative in case travel
around it is independent of path. For example, the first diagram commutes
iffJ = hg. If the second is commutative,
then in particular, travel from A to E is independent of path, whence
jgJ = ih.
A function J: A --+ B is injective (surjective) or is an injection (surjection)
2 Preliminaries
in case it has a left (right) inverse!':B --> A; that is, in casef'f = 1,.1 U!' = IB)
for some !,:B --> A. So (see (0.2))f:A --> B is injective (surjective) iff it is
one-to-one (onto B). A function f: A --> B is bijective or a bijection in case
it is both injective and surjective; that is, iff there exists a (necessarily unique)
inversef- I:B --> A withff-l = IB andf-lf= 1,.1.
If A ~ B, then the function i = i A,; B: A --> B defined by i = (IB I A): a f-+ a
for all a E A is called the inclusion map of A in B. Note that if A ~ Band
A ~ C, and if B =1= C, then iA,;B =1= iA,;c· Of course 1,.1 = (t';A'
With every pair (0,1) there is a Kronecker delta; that is, a function
b :(a, (3) f-+ b,p on the class of all ordered pairs defined by
I if a = [3
{
b,p = 0 if rx =1= [3.
Whenever we use a Kronecker delta, the context will make clear our choice
of the pair (0, 1).
0.2. The Axiom of Choice. Let A be a set, let Y} be a collection of non
empty subsets of B, and let a be a function from A to Y'. Then the Axiom of
Choice states that there is a function g: A --> B such that
g(a) E a(a) (a E A).
Suppose now thatf: B --> A is onto A; that is,/(B) = A. Then for each a E A,
there is a non-empty subset a(a) = f~( {a}) ~ B. Applying the Axiom of
Choice to A, the function a: af -+ a(a), and the collection Y' of subsets of B
prod uces a right inverse 9 for f, so as claimed in (O.l),f is surjective.
Let "'- be an equivalence relation on a set A. A subset R of A is a (comp/ete)
irredundant set of representatives of the relation '" in case for each a E A
there is a unique a(a) E R such that a '" a(a). The Axiom of Choice
guarantees the existence of such a set of representatives for each equivalence
relation.
0.3. Cartesian Products. A function a: A --> X will sometimes be called
an indexed set (in X indexed by A) or an A-tuple (in X) and will be written as
a = (X')'EA
where x, = a(rx). If A = {I, ... , n}, then we also use the standard variation
(X')'EA = (Xl"'" Xn)· Let (X')'EA be an indexed set of non-empty subsets of a
set X. Then the (cartesian) product of (Xal,EA is
XAX, = {a:A --> Xla(rx)EXa (ClEA)}.
That is, XAX, is just the set of all A-tuples (X,)'EA such that x, E Xa (rx E A).
By the Axiom of Choice XAX, is non-empty. If A = {1, ... , n}, then we
allow the notational variation
XA X, = XI X ... X Xn.
Note that if X = X, (a E A), then the cartesian product XAX, is simply XA,
the set of all functions from A to X. For each rx E A the Cl-projection