A characterisation of Morita algebras in terms of covers

Abstract

A pair $(A, P)$ is called a cover of $\operatorname{End}_A(P)^{op}$ if the Schur functor $\operatorname{Hom}_A(P, -)$ 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 $\operatorname{End}_A(P)^{op}$ is self-injective whenever $(A, P)$ is a cover of $\operatorname{End}_A(P)^{op}$ for a faithful projective-injective module $P$.