Rings and Categories of Modules / Edition 2

Rings and Categories of Modules / Edition 2

by Frank W. Anderson, Kent R. Fuller
     
 

ISBN-10: 0387978453

ISBN-13: 9780387978451

Pub. Date: 08/18/1998

Publisher: Springer New York

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.

…  See more details below

Overview

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.

Read More

Product Details

ISBN-13:
9780387978451
Publisher:
Springer New York
Publication date:
08/18/1998
Series:
Graduate Texts in Mathematics Series, #13
Edition description:
2nd ed. 1992
Pages:
378
Product dimensions:
6.20(w) x 9.40(h) x 1.10(d)

Table of Contents

Preface
0Preliminaries1
Ch. 1Rings, Modules and Homomorphisms10
1Review of Rings and their Homomorphisms10
2Modules and Submodules26
3Homomorphisms of Modules42
4Categories of Modules; Endomorphism Rings55
Ch. 2Direct Sums and Products65
5Direct Summands65
6Direct Sums and Products of Modules78
7Decomposition of Rings95
8Generating and Cogenerating105
Ch. 3Finiteness Conditions for Modules115
9Semisimple Modules - The Socle and the Radical115
10Finitely Generated and Finitely Cogenerated Modules - Chain Conditions123
11Modules with Composition Series133
12Indecomposable Decompositions of Modules140
Ch. 4Classical Ring-Structure Theorems150
13Semisimple Rings150
14The Density Theorem157
15The Radical of a Ring - Local Rings and Artinian Rings165
Ch. 5Functors Between Module Categories177
16The Hom Functors and Exactness - Projectivity and Injectivity178
17Projective Modules and Generators191
18Injective Modules and Cogenerators204
19The Tensor Functors and Flat Modules218
20Natural Transformations234
Ch. 6Equivalence and Duality for Module Categories250
21Equivalent Rings250
22The Morita Characterizations of Equivalence262
23Dualities269
24Morita Dualities278
Ch. 7Injective Modules, Projective Modules, and Their Decompositions288
25Injective Modules and Noetherian Rings - The Faith-Walker Theorems288
26Direct Sums of Countably Generated Modules - With Local Endomorphism Rings295
27Semiperfect Rings301
28Perfect Rings312
29Modules with Perfect Endomorphism Rings322
Ch. 8Classical Artinian Rings327
30Artinian Rings with Duality327
31Injective Projective Modules336
32Serial Rings345
Bibliography363
Index369

Read More

Customer Reviews

Average Review:

Write a Review

and post it to your social network

     

Most Helpful Customer Reviews

See all customer reviews >