On the endofiniteness of a key module over pure semisimple rings

Abstract

Let R R be a left pure semisimple ring such that there are no non-zero homomorphisms from preinjective modules to non-preinjective indecomposable modules in R R -mod, and let W W be the left key R R -module; i.e., W W is the direct sum of all non-isomorphic non-preinjective indecomposable direct summands of products of preinjective left R R -modules. We show that if the module W W is endofinite, then R R is a ring of finite representation type. This settles a question considered in [L. Angeleri Hügel, A key module over pure-semisimple hereditary rings, J. Algebra 307 (2007), 361-376] for hereditary rings.