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© 1973

Algebra

Rings, Modules and Categories I

Book

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

Table of contents

  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 83-109
    2. Carl Faith
      Pages 110-185
    3. Carl Faith
      Pages 230-300
    4. 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 388-401
    4. 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

About this book

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

  1. 1.Rutgers UniversityNew BrunswickUSA

Bibliographic information

  • Book Title Algebra
  • Book Subtitle Rings, Modules and Categories I
  • Authors Carl Faith
  • Series Title Die Grundlehren der mathematischen Wissenschaften
  • 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 Springer Book Archive
  • Hardcover ISBN 978-3-540-05551-8
  • Softcover ISBN 978-3-642-80636-0
  • eBook ISBN 978-3-642-80634-6
  • Series ISSN 0072-7830
  • Edition Number 1
  • Number of Pages XXIV, 568
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Topics Algebra
  • Buy this book on publisher's site