Purity in module categories and functor rings

The main goal of this course is to study the basic ideas and techniques that appear related to the study of pure-injectivity. We will introduce the idea of purity as a generalization of the concept of decomposition of modules into direct summands. Let us remark that this concept is basic in the modern study of the Representation Theory of Artin Algebras that has been developed by Auslander and Crawley-Boevey. Next, we define pure-injective and pure-projective modules. In order to study them, we introduce the fundamental concept of functor ring associated to a module category. Now, we obtain the basic properties of pure-injective modules, by means of this concept. Finally, we will outline some recent results and open problems related to purity. We will also apply our results to Representation Theory. In particular, we will discuss the pure-semisimple conjecture, as well as the role the infinite length modules play in the study of this theory.