Grundlehren der mathematischen Wissenschaften

Algebra

Rings, Modules and Categories I

Authors: Faith, Carl

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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 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 (18 chapters)

Buy this book

eBook $69.99
price for USA (gross)
valid through November 5, 2017
  • ISBN 978-3-642-80634-6
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $99.00
price for USA
  • ISBN 978-3-642-80636-0
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Algebra
Book Subtitle
Rings, Modules and Categories I
Authors
Series Title
Grundlehren der mathematischen Wissenschaften
Series Volume
190
Copyright
1973
Publisher
Springer-Verlag Berlin Heidelberg
Copyright Holder
Springer-Verlag, Berlin · Heidelberg
eBook ISBN
978-3-642-80634-6
DOI
10.1007/978-3-642-80634-6
Softcover ISBN
978-3-642-80636-0
Series ISSN
0072-7830
Edition Number
1
Number of Pages
XXIV, 568
Topics