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Book review

Author(s) Gabriel, P
Roiter, A.V.
Keller, B.
Title Representations of Finite-Dimensional Algebras
Publisher Springer-Verlag
Year of publication 1992
Reviewed by Andrei Marcus

Representation theory of finite dimensional algebras has developed very quickly in the last two decades and now is a mayor part of the theory of rings and modules. The aim of this excellent monograph written by three top researchers in the field is to supply the lack of oomprehensive textbooks. This lack constitutes a great impediment for beginners, so the authors put the stress on the foundations and, in order to reprove readability, they have included proofs of elementary statements, not easy to fiend elsewhere.

The book is divided into 14 sections. Sections 1-7 are introductory and accessible to graduate students, while Sections 8-14 give to the reader the possibility to attack the recent literature in the field. The topics discussed are centered around the finitely represented algebras, which have a reasonably complete theory, and some recent subjects are ommited. An account on modern trends in representation theory can be found in C.M. Ringel's survey "Recent Advances In the Representation Theory of Finite Dimonsional Algebras" in "Representation Theory of Finite Groups and Finite Dimensional Algebras" (Eds. G.O. Michler and C.M. Ringel), pp. 141-192.

The volume includes commented references after each section, a useful index and list of symbols and a large bibliography. The care in selecting a not yet well established terminology should be remarked.

For graduate studonte, for researchers and for anyone who is interested in noncommutative ring theory, this book is a real need and it will be a sound investment.