Rings and Their Modules

This book is an introduction to the theory of rings and modules that goes beyond what one normally obtains in a graduate course in abstract algebra. The theme of the text is the interplay between rings and modules. At times rings are investigated by considering given sets of conditions on the modules they admit and at other times rings of a certain type are considered to see what structure is forced on their modules. Standard topics in ring and module theory such as chain conditions on rings and modules, injective and projective modules and semisimple rings are included as well as subjects like category theory and homological algebra. The text also contains presentations on topics such as flat modules and coherent rings, injective envelopes, projective covers and perfect rings, reflexive modules and quasi-Frobenius rings, and graded rings and modules.

The book is a self-contained volume written in a very systematic style: all proofs are clear and easy for the reader to understand and all arguments are based on materials contained in the book. A problem sets follow each section.

It is assumed that the reader is familiar with concepts such as Zorn''s lemma, commutative diagrams and ordinal and cardinal numbers. It is also assumed that the reader has a basic knowledge of rings and their homomorphisms. The text is suitable for graduate and PhD students who have chosen ring theory for their research subject.

"This recent book is a welcome addition to the textbook literature on the subject of (mostly noncommutative) ring theory [...] The author of this text is clearly interested in making this material, and what follows, as accessible as possible to a graduate student. [...] This book has all the attributes of an excellent text. The writing is clear and reader-friendly, and there are both good examples and a reasonable number of exercises of varying difficulty. ... It is highly recommended, both as a text for ring theory courses at the various levels described above, or as an accessible resource for graduate students with an interest in pursuing research in ring theory."
Mark Hunacek, MAA Reviews