ON THE OSOFSKY–SMITH THEOREM

Abstract

AbstractWe recall a version of the Osofsky–Smith theorem in the context of a Grothendieck category and derive several consequences of this result. For example, it is deduced that every locally finitely generated Grothendieck category with a family of completely injective finitely generated generators is semi-simple. We also discuss the torsion-theoretic version of the classical Osofsky theorem which characterizes semi-simple rings as those rings whose every cyclic module is injective.