Rings and categories of modules

Frank W. Anderson, Kent R. Fuller

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 algebras.

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[目次]

  • x0. Preliminaries.- 1: Rings, Modules and Homomorphisms.- x1. Review of Rings and their Homomorphisms.- x2. Modules and Submodules.- x3. Homomorphisms of Modules.- x4. Categories of Modules
  • Endomorphism Rings.- 2: Direct Sums and Products.- x5. Direct Summands.- x6. Direct Sums and Products of Modules.- x7. Decomposition of Rings.- x8. Generating and Cogenerating.- 3: Finiteness Conditions for Modules.- x9. Semisimple Modules-The Sode and the Radical.- x10. Finitely Generated and Finitely Cogenerated Modules-Chain Conditions.- x11. Modules with Composition Series.- x12. Indecomposable Decompositions of Modules.- 4: Classical Ring-Structure Theorems.- x13. Semisimple Rings.- x14. The Density Theorem.- x15. The Radical of a Ring-Local Rings and Artinian Rings.- 5: Functors Between Module Categories.- x16. The Horn Functors and Exactness-Projectivity and Injectivity.- x17. Projective Modules and Generators.- x18. Injective Modules and Cogenerators.- x19. The Tensor Functors and Flat Modules.- x20. Natural Transformations.- 6: Equivalence and Duality for Module Categories.- x21. Equivalent Rings.- x22. The Morita Characterizations of Equivalence.- x23. Dualities.- x24. Morita Dualities.- 7: Injective Modules, Projective Modules, and Their Decompositions.- x25. Injective Modules and Noetherian Rings-The Faith-Walker Theorems.- x26. Direct Sums of Countably Generated Modules-With Local Endomorphism Rings.- x27. Semiperfect Rings.- x28. Perfect Rings.- x29. Modules with Perfect Endomorphism Rings.- 8: Classical Artinian Rings.- x30. Artinian Rings with Duality.- x31. Injective Projective Modules.- x32. Serial Rings.- References.

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この本の情報

書名 Rings and categories of modules
著作者等 Anderson, Frank W.
Fuller, Kent R.
Anderson Frank W.
シリーズ名 Graduate texts in mathematics
出版元 Springer-Verlag
刊行年月 c1992
版表示 2nd ed
ページ数 viii, 376 p.
大きさ 25 cm
ISBN 3540978453
0387978453
NCID BA18538214
※クリックでCiNii Booksを表示
言語 英語
出版国 アメリカ合衆国
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