A Course in Ring Theory

by Passman, Donald S.

• ISBN13: 9780821836804
• ISBN10: 0821836803
• Format: Hardcover
• Copyright: 2004-09-28
• Publisher: Chelsea Pub Co

Summary

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. The first part of the book, called "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 provedinclude 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.

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

Table of Contents provided by Publisher. All Rights Reserved.