Algebra: Rings, Modules and Categories I

ISBN-13: 9783642806360 / Angielski / Miękka / 2012 / 568 str.

Carl Faith

Opis

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 shot 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 corre 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."

Spis treści

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.