A generalization of quasi-Frobenius rings

Abstract

As a generalization of quasi-Frobenius rings, G. Azumaya [2] has investigated a ring with the property that every faithful left module is a generator' in the category of left mnodules, and he has proved that such a ring is left self-injective and a direct sum of indecomposable left ideals having minimal left ideals. In his proof, however, the existence of the faithful injective module plays an important role, while the injective module is not necessarily finitely generated,2 even if the ring is a left Artinian ring. Therefore, there naturally arises a problem: Is an Artinian ring quasi-Frobenius if every finitely generated faithful module is a generator? The purpose of this paper is to give an affirmative answer to this problem as a direct consequence from a more general result which is rather similar to that of Azumaya stated above and related to perfect rings introduced by H. Bass [3]. Throughout this paper we shall assume that the ring R has identity element 1 and all modules over it are unital. For a subset A of R we shall denote