Structure of rings (American mathematical society colloquium publications. Volume XXXVII)

Description

Since the appearance of the author's Theory of Rings and Artin, Nesbitt and Thrall's Rings with Minimum Condition, a number of important developments have taken place in the theory of (non-commutative) rings. These are: the structure theory of rings without finiteness assumptions, cohomology of algebras, and structure and representation theory of non-semi-simple rings (Frobenius algebras, quasi-Frobenius rings). The main purpose of the present volume is to give an account of the first of these developments. The tools which have been devised for the study of general rings yield improved proofs of the older structure results on rings with minimum condition and of finite dimensional algebras. We have therefore considered the specialization of the general results and methods to these classical cases. Thus the present volume includes virtually all the results on semi-simple rings which can be found in the two books cited before. For example, the theory of centralizers of finite dimensional simple subalgebras of simple rings with minimum condition appears as a special case of the Galois theory of the complete ring of linear transformations of a vector space over a division ling. We believe that the passage to the more general case gives a better insight into these results...