Springer Exercises in Modules and Rings (Problem Books in Mathematics)

This volume offers a compendium of exercises of varying degree of difficulty in the theory of modules and rings. Each section begins with an introduction giving the general background and the theoretical basis for the problems that follow. Review From the reviews: "This volume, a companion for the author’s graduate-level textbook … consists of a complete set of solutions to all … . should be on the bookshelf of every serious graduate student in theoretical mathematics. As one expects from the author, the writing in this book is clear and precise, and some of the problem solutions (and counterexamples to conjectures) presented are quite elegant." (Jonathan Golan, Mathematical Reviews, Issue 2007 h) "Exercises in Classical Ring Theory is an outgrowth of the author’s lectures on noncommutative rings given at Berkeley. The book presents solutions to over 400 exercises … . Those who purchase the book should find it helpful in the problem solving process as well as a demonstration of the different applications of theorems from ring theory." (Paul E. Bland, Zentralblatt MATH, Vol. 1121 (23), 2007) From the Back Cover For the Backcover This Problem Book offers a compendium of 639 exercises of varying degrees of difficulty in the subject of modules and rings at the graduate level. The material covered includes projective, injective, and flat modules, homological and uniform dimensions, noncommutative localizations and Goldie’s theorems, maximal rings of quotients, Frobenius and quasi-Frobenius rings, as well as Morita’s classical theory of category dualities and equivalences. Each of the nineteen sections begins with an introduction giving the general background and the theoretical basis for the problems that follow. All exercises are solved in full detail; many are accompanied by pertinent historical and bibliographical information, or a commentary on possible improvements, generalizations, and latent connections to other problems. This volume is designed as a problem book for the author’s Lectures on Modules and Rings (Springer GTM, Vol. 189), from which the majority of the exercises were taken. Some forty new exercises have been added to further broaden the coverage. As a result, this book is ideal both as a companion volume to Lectures, and as a source for independent study. For students and researchers alike, this book will also serve as a handy reference for a copious amount of information in algebra and ring theory otherwise unavailable from textbooks. An outgrowth of the author’s lecture courses and seminars over the years at the University of California at Berkeley, this book and its predecessor Exercises in Classical Ring Theory (Springer, 2003) offer to the mathematics community the fullest and most comprehensive reference to date for problem solving in the theory of modules and rings.