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AuthorFaith, Carl. author
TitleAlgebra [electronic resource] : Rings, Modules and Categories I / by Carl Faith
ImprintBerlin, Heidelberg : Springer Berlin Heidelberg, 1973
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Descript XXIV, 568 p. online resource


VI of Oregon lectures in 1962, Bass gave simplified proofs of a number of "Morita Theorems", incorporating ideas of Chase and Schanuel. One of the Morita theorems characterizes when there is an equivalence of categories mod-A R::! mod-B for two rings A and B. Morita's solution organizes ideas so efficiently that the classical Wedderburn-Artin theorem is a simple consequence, and moreover, a similarity class [AJ in the Brauer group Br(k) of Azumaya algebras over a commutative ring k consists of all algebras B such that the corresponding categories mod-A and mod-B consisting of k-linear morphisms are equivalent by a k-linear functor. (For fields, Br(k) consists of similarity classes of simple central algebras, and for arbitrary commutative k, this is subsumed under the Azumaya [51]1 and Auslander-Goldman [60J Brauer group. ) Numerous other instances of a wedding of ring theory and category (albeit a shotยญ gun wedding!) are contained in the text. Furthermore, in. my attempt to further simplify proofs, notably to eliminate the need for tensor products in Bass's exposition, I uncovered a vein of ideas and new theorems lying wholely within ring theory. This constitutes much of Chapter 4 -the Morita theorem is Theorem 4. 29-and the basis for it is a correยญ spondence theorem for projective modules (Theorem 4. 7) suggested by the Morita context. As a by-product, this provides foundation for a rather complete theory of simple Noetherian rings-but more about this in the introduction


to Volume -- Foreword on Set Theory -- I Introduction to the Operations: Monoid, Semigroup, Group, Category, Ring, and Module -- 1. Operations: Monoid, Semigroup, Group, and Category -- 2. Product and Coproduct -- 3. Ring and Module -- 4. Correspondence Theorems for Projective Modules and the Structure of Simple Noetherian Rings -- 5. Limits, Adjoints, and Algebras -- 6. Abelian Categories -- II Structure of Noetherian Semiprime Rings -- 7. General Wedderburn Theorems -- 8. Semisimple Modules and Homological Dimension -- 9. Noetherian Semiprime Rings -- 10. Orders in Semilocal Matrix Rings -- III Tensor Algebra -- 11. Tensor Products and Flat Modules -- 12. Morita Theorems and the Picard Group -- 13. Algebras over Fields -- IV Structure of Abelian Categories -- 14. Grothendieck Categories -- 15. Quotient Categories and Localizing Functors -- 16. Torsion Theories, Radicals, and Idempotent, Topologizing, and Multiplicative Sets

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