Endofinite modules and pure semisimple rings

Let R be a right pure semisimple ring, i.e., a ring R such that every right R-module is a direct sum of finitely generated modules. It is proved that R is of finite representation type if and only if every finitely presented (indecomposable) right R-module is endofinite, if and only if every finitely presented right R-module has a left artinian endomorphism ring. As applications, we obtain an alternative proof of the pure semisimplicity conjecture for PI-rings, and new criteria for a right pure semisimple ring to be of finite representation type.