Abstract
In this paper, we generalize the concept of direct-injective modules to fi nite-direct-injective modules. Various basic properties of these modules are studied. We show that the class of fi nite-direct-injective modules lies between the class of direct-injective modules and the class of simple-direct-injective modules. Also, we characterize semisimple artinian rings, V -rings and regular rings in terms of fi nite-direct-injective modules.
Highlights
Throughout this paper, all rings are associative rings with unity and all modules are unitary right modules
The notations N ≤ M, N ≤ess M and N ≤ M means that N is a submodule, an essential submodule and a direct summand of M, respectively
A module M is called simple-direct-injective if every simple submodule isomorphic to a direct summand of M is itself a direct summand of M
Summary
Throughout this paper, all rings are associative rings with unity and all modules are unitary right modules. A module M is called simple-direct-injective if every simple submodule isomorphic to a direct summand of M is itself a direct summand of M. A module M is called finite-direct-injective if every finitely generated submodule of M isomorphic to a direct summand of M is itself a direct summand of M. It is the generalization of direct-injective modules. We give an example of a finite-directinjective module that is not a direct-injective module. We characterize rings R for which every singular right R-module is finite-directinjective
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