Rings and Categories of Modules
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...
| Main Authors: | , |
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| Format: | eBook |
| Language: | English |
| Published: |
New York, NY
Springer New York
1992, 1992
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| Edition: | 2nd ed. 1992 |
| Series: | Graduate Texts in Mathematics
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| Subjects: | |
| Online Access: | |
| Collection: | Springer Book Archives -2004 - Collection details see MPG.ReNa |
Table of Contents:
- §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