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 ≅ B and B ⊆ ⊕ M , we have A ⊆ ⊕ 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] . [ABSTRACT FROM AUTHOR]