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Groups, Rings, Modules

Maurice Auslander and David A. Buchsbaum
Dover Publications
Publication Date: 
Number of Pages: 
[Reviewed by
Allen Stenger
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This is primarily a rings and modules book, and is primarily about using the structure of a ring’s modules to infer the structure of the ring and vice versa. It also discusses groups and fields, but in much less detail. The book uses category language throughout. It assumes few prerequisites, and is positioned as a second-year abstract algebra course. It has good explanations and is easy to follow. The present book is a 2014 unaltered reprint of the 1974 Harper & Row edition.

The book’s big weakness is that is skimpy on examples. It does have a large number of exercises, most of them challenging, but even there most of them are to prove additional results that did not fit into the main exposition, and few examples are given.

Although the book is a valuable reference, I think it does not work well as a second course, because of the lack of examples and because it has much more information about rings and modules than anyone at that level would want to know. I think a better choice would be one of the more balanced abstract algebra textbooks, such as Hungerford’s Abstract Algebra: An Introduction (although it does not cover modules), or Artin’s Algebra, or at a more advanced level Dummit & Foote’s Abstract Algebra, which has very thorough coverage of modules that is well integrated with the rest of the text.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.

Sets and Maps
Monoids and Groups
Unique Factorization Domains
General Module Theory
Semisimple Rings and Modules
Artinian Rings
Localization and Tensor Products
Principal Ideal Domains
Applications of the Fundamental Theorem
Algebraic Field Extensions
Dedekind Domains