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.