LAUSR

The notion of simple-direct-injective modules which are a general- ization of injective modules unifies C2 and C3-modules. In the present paper, we introduce the notion of the semisimple-direct-injective module which gives a unified viewpoint of C2, C3, SSP properties and simple-direct-injective mod- ules. It is proved that a ring R is Artinian serial with the Jacobson radical square zero if and only if every semisimple-direct-injective right R-module has the SSP and, for any family of simple injective right R-modules {S_i }_I , ⊕_I S_i is injective. We also show that R is a right Noetherain right V-ring if and only if every right R-module has a semisimple-direct-injective-envelope if and only if every right R-module has a semisimple-direct-injective-cover.