Preinjective modules over pure semisimple rings

For a left pure semisimple ring RR, it is shown that the local duality establishes a bijection between the preinjective left RR-modules and the preprojective right RR-modules, and any preinjective left RR-module is the source of a left almost split morphism. Moreover, if there are no nonzero homomorphisms from preinjective modules to non-preinjective indecomposable modules in RR-mod, the direct sum of all non-preinjective indecomposable direct summands of products of preinjective left RR-modules is a finitely generated product-complete module. This generalizes a recent theorem of Angeleri Hügel [L. Angeleri Hügel, A key module over pure-semisimple hereditary rings, J. Algebra 307 (2007) 361–376] for hereditary rings.