Rings whose quasi-injective modules are injective

Abstract:

A ring R is called a V-ring, respectively SSI-ring, respectively QII-ring if simple, respectively semisimple, respectively quasi-injective, right R-modules are injective. We show that R is SSI if and only if R is a right noetherian V-ring and that any SSI-ring is a finite ring direct sum of simple SSI-rings. We show that if R is left noetherian and SSI then R is QII provided R is hereditary and that in order for R to be hereditary it suffices that maximal right ideals of R be reflexive. An example of Cozzens is cited to show these rings need not be artinian.