On a class of QI-rings

Abstract

The concept of weak relative-injectivity of modules was introduced originally in [10], where it is shown that a semiperfect ring R is such that every cyclic right module is embeddable essentially in a projective right R -module if and only if R is right artinian and every indecomposable projective right R -module is uniform and weakly R -injective. We show that in the above characterization the requirement that indecomposable projective right R -modules be uniform is superfluous (Theorem 1.11). In this paper we further the study of weak relative-injectivity by considering the class of rings for which every right module is weakly injective relative to every finitely generated right module. We refer to such rings as right weakly-semisimple rings. The class of right weakly-semisimple rings includes properly all semisimple rings and is a subclass of the class of right QI-rings. A ring R is said to be a right QI-ring if every quasi-injective right R -module is injective. QI-rings have been studied in [2], [3], [4], [6], [7], [8], [11], among others.