This monograph provides a thorough treatment of module theory, a subfield of algebra.

The authors develop an approximation theory as well as realization theorems and present some of its recent applications, notably to infinite-dimensional combinatorics and model theory.

The book is devoted to graduate students interested in algebra as well as to experts in module theory.

This second, revised and substantially extended edition of Approximations and Endomorphism Algebras of Modules reflects both the depth and the width of recent developments in the area since the first edition appeared in 2006. The new division of the monograph into two volumes roughly corresponds to its two central topics, approximation theory (Volume 1) and realization theorems for modules (Volume 2).

It is a widely accepted fact that the category of all modules over a general associative ring is too complex to admit classification. Unless the ring is of finite...