A characterisation of Morita algebras in terms of covers

Abstract

A pair (A, P ) is called a cover of EndA(P ) op if the Schur functor HomA(P, −) is fully faithful on the full subcategory of projective A-modules, for a given projective A-module P . By definition, Morita algebras are the covers of self-injective algebras and then P is a faithful projective-injective module. Conversely, we show that A is a Morita algebra and EndA(P ) op is self-injective whenever (A, P ) is a cover of EndA(P ) op for a faithful projective-injective module P .