When proper cyclics are homomorphic image of injectives

Abstract

Quasi-Frobenius rings are precisely rings over which any right module is a homomorphic image of an injective module. We investigate the structure of rings whose proper cyclic right modules are homomorphic image of injectives. The class of such rings properly contains that of right self-injective rings. We obtain some structure theorems for rings satisfying the said property and apply them to the Artin algebra case: It follows that an Artin algebra with this property is Quasi-Frobenius.