On rings with cyclic almost-injective modules

Abstract

It is shown that every finitely generated right [Formula: see text]-module is almost injective if and only if every cyclic right [Formula: see text]-module is almost injective, if and only if [Formula: see text] is a right [Formula: see text]-ring with [Formula: see text] and there is a finite set of orthogonal idempotents [Formula: see text] in [Formula: see text] such that [Formula: see text] is an injective local right [Formula: see text]-module of length two for every [Formula: see text] and [Formula: see text].