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A Course in Ring Theory
Donald S. Passman, University of Wisconsin, Madison, WI
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AMS Chelsea Publishing
2004; 306 pp; hardcover
Volume: 348
ISBN-10: 0-8218-3680-3
ISBN-13: 978-0-8218-3680-4
List Price: US$49
Member Price: US$44.10
Order Code: CHEL/348.H
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First published in 1991, this book contains the core material for an undergraduate first course in ring theory. Using the underlying theme of projective and injective modules, the author touches upon various aspects of commutative and noncommutative ring theory. In particular, a number of major results are highlighted and proved.

Part I, "Projective Modules", begins with basic module theory and then proceeds to surveying various special classes of rings (Wedderburn, Artinian and Noetherian rings, hereditary rings, Dedekind domains, etc.). This part concludes with an introduction and discussion of the concepts of the projective dimension.

Part II, "Polynomial Rings", studies these rings in a mildly noncommutative setting. Some of the results proved include the Hilbert Syzygy Theorem (in the commutative case) and the Hilbert Nullstellensatz (for almost commutative rings).

Part III, "Injective Modules", includes, in particular, various notions of the ring of quotients, the Goldie Theorems, and the characterization of the injective modules over Noetherian rings.

The book contains numerous exercises and a list of suggested additional reading. It is suitable for graduate students and researchers interested in ring theory.

Readership

Graduate students and research mathematicians interested in ring theory.

Reviews

""There seems to be an emerging consensus as to what material should constitute the core of a first course in module-theoretic ring theory ... The book ... is definitely within the bounds of that consensus ... presentation is clear, the proofs are often quite ingenious and the exercises are well chosen ... definitely suitable for use as a textbook.""

-- Mathematical Reviews

"This excellently written book, which has been published originally by Wadsworthand Brooks in 1991, is already a classic ... A book recommendable now as before!"

-- Monatshefte für Mathematik

Table of Contents

Projective modules
  • Modules and homomorphisms
  • Projective modules
  • Completely reducible modules
  • Wedderburn rings
  • Artinian rings
  • Hereditary rings
  • Dedekind domains
  • Projective dimension
  • Tensor products
  • Local rings
Polynomial rings
  • Skew polynomial rings
  • Grothendieck groups
  • Graded rings and modules
  • Induced modules
  • Syzygy theorem
  • Patching theorem
  • Serre conjecture
  • Big projectives
  • Generic flatness
  • Nullstellensatz
Injective modules
  • Injective modules
  • Injective dimension
  • Essential extensions
  • Maximal ring of quotients
  • Classical ring of quotients
  • Goldie rings
  • Uniform dimension
  • Uniform injective modules
  • Reduced rank
  • Index
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