Rings and categories of modules / Frank W. Anderson, Kent R. Fuller

A very clearly written exposition of the general theory of rings and their modules. First, a thumbnail sketch of what the book contains. Despite the title, the formal categorical aspects of the subject do not intrude on the author's love affair with classical noncommutative ring and module theory, extending through Chapter 4, which includes the Chevalley-Jacobson density theorem for semisimple modules. (Polynomial rings are relegated to exercises!) The Wedderburn-Artin theorem for rings with the descending chain condition are familiar corollaries. The Jacobson radical is defined, and characterized [à la Perlis] as the maximal quasi-regular one-sided ideal. The more modern aspects of the subject—the homological and categorical—begin in Chapter 5 with the hom and tensor functors, the study of their exactness, natural transformations of functors, and various theorems on adjoint functors. This sets the stage for the study of equivalence of categories in Chapter 6, notably, Morita's characterization of when the two categories of all modules mod-A and mod-B over two different rings A and B are equivalent. This constitutes the first essential use of category theory in the text, and the authors show that mod-A is determined up to equivalence by the full subcategory of finitely generated modules. Morita duality is developed in an essentially dual, but necessarily more arduous, manner. (Zentralblatt)

Bibliogr. p. 327-331. Index