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Algebra

Rings, Modules and Categories I

  • Book
  • © 1973

Overview

Authors:
  1. Carl Faith

    1. Rutgers University, New Brunswick, USA

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

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Table of contents (18 chapters)

  1. Front Matter

    Pages I-XXIII
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  2. Introduction to Volume I

    1. Introduction to Volume I

      • Carl Faith
      Pages 1-1

  3. Foreword on Set Theory

    1. Foreword on Set Theory

      • Carl Faith
      Pages 2-42

  4. Introduction to the Operations: Monoid, Semigroup, Group, Category, Ring, and Module

    1. Operations: Monoid, Semigroup, Group, and Category

      • Carl Faith
      Pages 43-82

    2. Product and Coproduct

      • Carl Faith
      Pages 83-109

    3. Ring and Module

      • Carl Faith
      Pages 110-185

    4. Correspondence Theorems for Projective Modules and the Structure of Simple Noetherian Rings

      • Carl Faith
      Pages 185-229

    5. Limits, Adjoints, and Algebras

      • Carl Faith
      Pages 230-300

    6. Abelian Categories

      • Carl Faith
      Pages 300-321

  5. Structure of Noetherian Semiprime Rings

    1. Front Matter

      Pages 322-324
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    2. General Wedderburn Theorems

      • Carl Faith
      Pages 325-365

    3. Semisimple Modules and Homological Dimension

      • Carl Faith
      Pages 365-388

    4. Noetherian Semiprime Rings

      • Carl Faith
      Pages 388-401

    5. Orders in Semilocal Matrix Rings

      • Carl Faith
      Pages 401-417

  6. Tensor Algebra

    1. Front Matter

      Pages 418-419
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    2. Tensor Products and Flat Modules

      • Carl Faith
      Pages 419-442

    3. Morita Theorems and the Picard Group

      • Carl Faith
      Pages 443-459

    4. Algebras over Fields

      • Carl Faith
      Pages 460-483

  7. Structure of Abelian Categories

    1. Front Matter

      Pages 484-486
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    2. Grothendieck Categories

      • Carl Faith
      Pages 486-497
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Keywords

About this book

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.

Authors and Affiliations

  • Rutgers University, New Brunswick, USA

    Carl Faith

Bibliographic Information

  • Book Title: Algebra

  • Book Subtitle: Rings, Modules and Categories I

  • Authors: Carl Faith

  • Series Title: Grundlehren der mathematischen Wissenschaften

  • DOI: https://doi.org/10.1007/978-3-642-80634-6

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag, Berlin ยท Heidelberg 1973

  • Softcover ISBN: 978-3-642-80636-0Published: 03 August 2012

  • eBook ISBN: 978-3-642-80634-6Published: 06 December 2012

  • Series ISSN: 0072-7830

  • Series E-ISSN: 2196-9701

  • Edition Number: 1

  • Number of Pages: XXIV, 568

  • Topics: Algebra

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