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The Algebraic Structure of Group Rings

Donald S. Passman
Dover Publications
Publication Date: 
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Fernando Q. Gouvêa
, on

If G is a group and R is a commutative ring, the ring R[G] consists of all finite formal R-linear combinations of elements of G, given the "obvious" product, i.e., use the group law to multiply elements of G and enforce the usual ring properties. These rings play a fundamental role in the representation theory of G: if K is a field, a representation of G defined over K is "the same thing" as a K[G]-module.

Passman's 1977 book focused on group rings as rings, i.e., it is most interested in ring-theoretic questions, as opposed to questions of representation theory. It attempted to gather into one volume all of the knowledge then available about the structure of group rings. As such, it is basically a reference book, and a very useful one.

Passman updated the original edition in 1985, and this Dover reprint is based on that version. I have owned a copy of the 1977 Wiley edition since my days as an M.A. student in São Paulo, and I'll admit to being a little frustrated at not finding much information in this version about what was changed. I guess I'll have to keep both!

I had forgotten that introduction begins with "I hadn't meant this book to be so long. But once the ground rules were set up, I had no choice." But I hadn't forgotten a phrase from page viii:

Now for a word on notation. Unfortunately, the word is "inconsistent."

Anyone who has the guts and the humor to write that deserves a tip of my hat.

Fernando Q. Gouvêa played around with group rings when he was young. Now he is old, and has become the Carter Professor of Mathematics at Colby College and the editor of MAA Reviews.

Part 1 Introduction
1. Group Rings
2. The Trace Map
3. The Augmentation Ideal
Part 2 Linear Identities
4. The Center
5. Polynomial Identities
6. Bounded Representation Degree
7. The Semisimplicity Problem
8. The Nilpotent Radical
9. Primitive Rings
Part 3 Finiteness Properties
10. Chain Conditions
11. Nilpotent Groups
12. Finite Dimensional Modules
13. The Zero Divisor Problem
14. Isomorphism Questions