Preinjective modules and finite representation type of artinian rings

Abstract

Let R be a left artinian ring such that every finitely presented right .ft-module is of finite endolength. It is shown that the cardinality of the set of isomorphism classes of preinjective right R-modules is less than or equal to the cardinality of the set of isomorphism classes of preprojective left R-modules, and R is of finite representation type if and only if these cardinal numbers are finite and equal to each other. As a consequence, we deduce a theorem, due to Herzog [17], asserting that a left pure semisimple ring R is of finite representation type if and only if the number of non-isomorphic preinjective right R-modules is the same as the number of non-isomorphic preprojective left .R-modules. Further applications are also given to provide new criteria for artinian rings with self-duality and artinian Pi-rings to be of finite representation type, which imply in particular the validity of the pure semisimple conjecture for these classes of rings.