Exercises in Modules and RingsThe 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 math ematical 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, conclud ing with Morita's theory of category equivalences and dualities. |
Contents
Free Modules Projective and Injective Modules | 1 |
Exercises for 1 | 2 |
2 Projective Modules | 21 |
3 Injective Modules | 60 |
Exercises for 3 | 61 |
Flat Modules and Homological Dimensions | 97 |
5 Homological Dimensions | 131 |
More Theory of Modules | 155 |
12 Artinian Rings of Quotients | 260 |
More Rings of Quotients | 271 |
14 Martindals Ring of Quotients | 287 |
Frobenius and QuasiFrobenius Rings | 299 |
16 Frobenius Rings and Symmetric Algebras | 315 |
Matrix Rings Categories of Modules and Morita Theory | 343 |
Exercises for 17 | 344 |
18 Morita Theory of Category Equivalences | 353 |
7 Singular Submodules and Nonsingular Rings | 185 |
8 Dense Submodules and Rationa1 Hu11s | 205 |
Rings of Quotients | 217 |
10 Classical Ring of Quotients | 222 |
11 Right Goldie Rings and Goldies Theorem | 242 |
19 Morita Duality Theory | 378 |
| 403 | |
| 407 | |
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Common terms and phrases
0-divisor abelian group algebra ann(m artinian ring assume automorphism cogenerator commutative ring consider contains Conversely Dedekind-finite defined desired direct sum direct summand division ring domain embedded endomorphism equivalent exact sequence exists f.cog f.g. projective fact flat follows free module Frobenius k-algebra functor hence homomorphism idempotent implies indecomposable injective hull injective module isomorphism k-algebra Kasch last exercise left ideal left R-module Lemma M₁ Math matrix ring maximal ideal minimal Mn(R multiplication Neumann regular ring noetherian ring nonzero element P₁ permutation prime ideal projective module proof prove QF ring Qmax R-homomorphism R-submodule regular elements resp Rickart right annihilators right artinian right Goldie right ideal right noetherian right nonsingular right R-module ring of quotients satisfies semiprime semisimple simple soc(RR Solution split stably finite submodule surjection Theorem u.dim zero