Algebra: Rings, Modules and Categories I

✍ Scribed by Carl Faith (auth.)

• Publisher: Springer-Verlag Berlin Heidelberg
• Year: 1973
• Tongue: English
• Leaves: 589
• Series: Die Grundlehren der mathematischen Wissenschaften 190
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Front Matter....Pages I-XXIII
Introduction to Volume I....Pages 1-1
Foreword on Set Theory....Pages 2-42
Operations: Monoid, Semigroup, Group, and Category....Pages 43-82
Product and Coproduct....Pages 83-109
Ring and Module....Pages 110-185
Correspondence Theorems for Projective Modules and the Structure of Simple Noetherian Rings....Pages 185-229
Limits, Adjoints, and Algebras....Pages 230-300
Abelian Categories....Pages 300-321
Front Matter....Pages 322-324
General Wedderburn Theorems....Pages 325-365
Semisimple Modules and Homological Dimension....Pages 365-388
Noetherian Semiprime Rings....Pages 388-401
Orders in Semilocal Matrix Rings....Pages 401-417
Front Matter....Pages 418-419
Tensor Products and Flat Modules....Pages 419-442
Morita Theorems and the Picard Group....Pages 443-459
Algebras over Fields....Pages 460-483
Front Matter....Pages 484-486
Grothendieck Categories....Pages 486-497
Quotient Categories and Localizing Functors....Pages 498-519
Torsion Theories, Radicals, and Idempotent, Topologizing, and Multiplicative Sets....Pages 519-537
Back Matter....Pages 538-567

✦ Subjects

Algebra