Rings and Categories of Modules

Rings and Categories of Modules

Book Description:

This book is intended to provide a self-contained account of much of the theory of rings and modules. The theme of the text throughout is the relationship between the one-sided ideal structure a ring may possess and the behavior of its categories of modules. Following a brief outline of the foundations, the book begins with the basic definitions and properties of rings, modules and homomorphisms. The remainder of the text gives comprehensive treatments of direct sums, finiteness conditions, the Wedderburn-Artin Theorem, the Jacobson radical, the hom and tensor functions, Morita equivalence and duality, decomposition theory, and semiperfect and perfect rings. This second edition includes a chapter containing many of the classical results on Artinian rings that have helped form the foundation for much of contemporary research on the representation theory of Artinian rings and finite-dimensional algebra

0. Preliminaries — 1: Rings, Modules and Homomorphisms
1. Review of Rings and their Homomorphisms
2. Modules and Submodules
3. Homomorphisms of Modules
4. Categories of Modules; Endomorphism Rings — 2: Direct Sums and Products
5. Direct Summands
6. Direct Sums and Products of Modules
7. Decomposition of Rings
8. Generating and Cogenerating — 3: Finiteness Conditions for Modules
9. Semisimple Modules — The Sode and the Radical
10. Finitely Generated and Finitely Cogenerated Modules — Chain Conditions
11. Modules with Composition Series
12. Indecomposable Decompositions of Modules — 4: Classical Ring-Structure Theorems
13. Semisimple Rings
14. The Density Theorem
15. The Radical of a Ring — Local Rings and Artinian Rings — 5: Functors Between Module Categories
16. The Horn Functors and Exactness — Projectivity and Injectivity
17. Projective Modules and Generators
18. Injective Modules and Cogenerators
19. The Tensor Functors and Flat Modules
20. Natural Transformations — 6: Equivalence and Duality for Module Categories
21. Equivalent Rings
22. The Morita Characterizations of Equivalence
23. Dualities
24. Morita Dualities — 7: Injective Modules, Projective Modules, and Their Decompositions
25. Injective Modules and Noetherian Rings — The Faith-Walker Theorems
26. Direct Sums of Countably Generated Modules — With Local Endomorphism Rings
27. Semiperfect Rings
28. Perfect Rings
29. Modules with Perfect Endomorphism Rings — 8: Classical Artinian Rings
30. Artinian Rings with Duality
31. Injective Projective Modules
32. Serial Rings — References.

 

Rings and Categories of Modules

Author(s): Frank W. Anderson, Kent R. Fuller (auth.)

Series: Graduate Texts in Mathematics 13

Publisher: Springer-Verlag New York, Year: 1992

ISBN: 978-1-4612-8763-6

[PDF] Rings and Categories of Modules Table Of Contents

Front Matter….Pages i-ix
Preliminaries….Pages 1-9
Rings, Modules and Homomorphisms….Pages 10-64
Direct Sums and Products….Pages 65-114
Finiteness Conditions for Modules….Pages 115-149
Classical Ring-Structure Theorems….Pages 150-176
Functors Between Module Categories….Pages 177-249
Equivalence and Duality for Module Categories….Pages 250-287
Injective Modules, Projective Modules, and Their Decompositions….Pages 288-326
Classical Artinian Rings….Pages 327-361
Back Matter….Pages 363-378


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