Problem Books in Mathematics. New York, NY: Springer (ISBN 0-387-98850-5/hbk). xviii, 412 p. EUR 46.95/net; SFR 83.00; $ 59.95; £ 36.00 (2007).
Exercises in Classical Ring Theory is an outgrowth of the author’s lectures on noncommutative rings given at Berkeley. The book presents solutions to over 400 exercises, of varying degrees of difficulty, from the second edition of the author’s book A First Course in Noncommutative Rings (2001; Zbl 0980.16001
). The problems cover topics such as the Wedderburn-Artin theory of semisimple rings, the Jacobson radical, prime and semiprime rings, primitive and semiprimitive rings, representations of groups and algebras, division rings, ordered rings, local rings, and semiperfect and perfect rings.
The author states in the preface that he envisions that the book will be helpful in at least three ways: (1) as a companion to First Course (second edition), (2) as a source book for self-study in problem solving, and (3) as a convenient reference for much of the folklore in classical ring theory that is not available elsewhere. In this reviewer’s opinion, the book will more than meet these goals. Those who purchase the book should find it helpful in the problem solving process as well as a demonstration of the different applications of theorems from ring theory.