When are Quasi-Injectives Injective?

Abstract

We call a ring R (associative and with identity) for which every quasi-injective right R-module is injective a QII-ring. Similarly R is called an SSI-ring when every semisimple right R-module is injective. Clearly every semisimple, artinian ring is a QII-ring and every QII-ring is an SSI-ring. One then asks whether these inclusions among classes of rings are proper. The purpose of this note is to point out an instance when SSI implies QII It is then easy to see that an example of Cozzens shows that the class of QII-rings properly contains the class of semisimple, artinian rings.