Algebras, Rings and Modules (eBook)

The text of the first volume of the book covers the major topics in ring and module theory and includes both fundamental classical results and more recent developments. The basic tools of investigation are methods from the theory of modules, which allow a very simple and clear approach both to classical and new results. An unusual main feature of this book is the use of the technique of quivers for studying the structure of rings. A considerable part of the first volume of the book is devoted to a study of special classes of rings and algebras, such as serial rings, hereditary rings, semidistributive rings and tiled orders. Many results of this text until now have been available in journal articles only. This book is aimed at graduate and post-graduate students and for all mathematicians who use algebraic techniques in their work. This is a self-contained book which is intended to be a modern textbook on the structure theory of associative rings and algebras and is suitable for independent study. TOC:Preface.- 1 Preliminaries.- 2 Decompositions of Rings.- 3 Artinian and Noetherian Rings.- 4 Categories and Functors.- 5 Projectives, Injectives and Flats.- 6 Homological Dimensions.- 7 Integral Domains.- 8 Dedekind Domains.- 9 Goldie Rings.- 10 Semiperfect Rings.- 11 Quivers of Rings.- 12 Serial Rings and Modules.- 13 Serial Rings and Their Properties.- 14 Semiperfect Semidistributive Rings.- Suggestions for further reading.- Index.- Name Index.