Modules whose injectivity domains are restricted to semi-artinian modules

Abstract

We introduce the notion of semi-poor modules and consider the possibility that all modules are either injective or semi-poor. This notion gives a generalization of poor modules that have minimal injectivity domain. A module [Formula: see text] is called semi-poor if whenever it is [Formula: see text]-injective and [Formula: see text], then the module [Formula: see text] has nonzero socle. In this paper the properties of semi-poor modules are investigated and are used to characterize various families of rings. We introduce the rings over which every module is either semi-poor or injective and call such condition property [Formula: see text]. The structure of the rings that have the property [Formula: see text] is completely determined. Also, we give some characterizations of rings with the property [Formula: see text] in the language of the lattice of hereditary pretorsion classes over a given ring. It is proved that a ring [Formula: see text] has the property [Formula: see text] iff either [Formula: see text] is right semi-Artinian or [Formula: see text] where [Formula: see text] is a semisimple Artinian ring and [Formula: see text] is right strongly prime and a right [Formula: see text]-ring with zero right socle.