Exercises in Modules and Rings

✍ Scribed by T. Y. Lam (auth.)

• Publisher: Springer-Verlag New York
• Year: 2007
• Tongue: English
• Leaves: 426
• Series: Problem Books in Mathematics
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Front Matter....Pages i-xviii
Free Modules, Projective, and Injective Modules....Pages 1-95
Flat Modules and Homological Dimensions....Pages 97-154
More Theory of Modules....Pages 155-216
Rings of Quotients....Pages 217-269
More Rings of Quotients....Pages 271-298
Frobenius and Quasi-Frobenius Rings....Pages 299-342
Matrix Rings, Categories of Modules and Morita Theory....Pages 343-402
Back Matter....Pages 403-413

✦ Subjects

Algebra; Associative Rings and Algebras