Rings with finite essential socle

Abstract

Let R R be a ring such that every direct summand of the injective envelope E = E ( R R ) E=E(R_R) has an essential finitely generated projective submodule. We show that, if the cardinal of the set of isomorphism classes of simple right R R -modules is no larger than that of the isomorphism classes of minimal right ideals, then R R R_R cogenerates the simple right R R -modules and has finite essential socle. This extends Osofsky’s theorem which asserts that a right injective cogenerator ring has finite essential right socle. It follows from our result that if R R R_R is a CS cogenerator, then R R R_R is already an injective cogenerator and, more generally, that if R R R_R is CS and cogenerates the simple right R R -modules, then it has finite essential socle. We show with an example that in the latter case R R R_R need not be an injective cogenerator.