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Book review

Author(s) Nastasescu, C .
Oystaeyen, F. van
Title Dimensions of Ring Theory
Publisher D.Reidel Publishing Company
Year of publication 1987
Reviewed by Mirela Stefanescu

The notions of dimensions are more and more used in modern ring theory. The authors, two specialists in this domain, have managed to unify different definitions of dimensions, by using a lattice theoretical point of view, an old Nastasescu's idea. Therefore, the first chapter is devoted to study the finiteness conditions for lattices, while the second chapter presents immediate applications to modules and rings.

The presentation of some topics as Artinian rings, Noetherian rings, Goldi's theorems, other generalities on modules is the preparation for the following chapters. The authors study Krull dimension, Gabriel dimension, relative Krull and Gabriel dimensions, homological dimension, global dimension and some classes of rings and modules of finite dimensions.

In the end of the book, they present Gelfand-Kirillov dimension, the only dimension which cannot be treated in the lattice theoretical manner.

Examples of rings and modules with specific dimensions, the structure of such rings, applications to other topics in algebra make the monography very interesting for students and researchers in ring and module theory.