Simple-direct-injective modules

A module M over a ring is called simple-direct-injective if, whenever A and B are simple submodules of M with A congruent to B and B subset of(circle plus) M, we have A subset of(circle plus) M. Various basic properties of these modules are proved, and some well-studied rings are characterized using simple-direct-injective modules. For instance, it is proved that a ring R is artinian serial with Jacobson radical square zero if and only if every simple-direct-injective right R-module is a C3-module, and that a regular ring R is a right V-ring (i.e., every simple right R-module is injective) if and only if every cyclic right R-module is simple-direct-injective. The latter is a new answer to Fisher's question of when regular rings are V-rings [8]. (C) 2014 Elsevier Inc. All rights reserved.