Algebra

Rings, Modules and Categories I

• Carl Faith
Book

Part of the Die Grundlehren der mathematischen Wissenschaften book series (GL, volume 190)

1. Front Matter
Pages I-XXIII
2. Introduction to Volume I

1. Carl Faith
Pages 1-1
3. Foreword on Set Theory

1. Carl Faith
Pages 2-42
4. Introduction to the Operations: Monoid, Semigroup, Group, Category, Ring, and Module

1. Carl Faith
Pages 43-82
2. Carl Faith
Pages 83-109
3. Carl Faith
Pages 110-185
4. Carl Faith
Pages 185-229
5. Carl Faith
Pages 230-300
6. Carl Faith
Pages 300-321
5. Structure of Noetherian Semiprime Rings

1. Front Matter
Pages 322-324
2. Carl Faith
Pages 325-365
3. Carl Faith
Pages 365-388
4. Carl Faith
Pages 388-401
5. Carl Faith
Pages 401-417
6. Tensor Algebra

1. Front Matter
Pages 418-419
2. Carl Faith
Pages 419-442
3. Carl Faith
Pages 443-459
4. Carl Faith
Pages 460-483
7. Structure of Abelian Categories

1. Front Matter
Pages 484-486
2. Carl Faith
Pages 486-497

Introduction

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.

Keywords

Autodesk Maya Coproduct Kategorie Modul algebra character class commutative ring group matrix ring ring theory semigroup theorem torsion

Authors and affiliations

• Carl Faith
• 1
1. 1.Rutgers UniversityNew BrunswickUSA

Bibliographic information

• DOI https://doi.org/10.1007/978-3-642-80634-6
• Copyright Information Springer-Verlag Berlin Heidelberg 1973
• Publisher Name Springer, Berlin, Heidelberg
• eBook Packages
• Print ISBN 978-3-642-80636-0
• Online ISBN 978-3-642-80634-6
• Series Print ISSN 0072-7830