On generalizations of injective modules

As a proper generalization of injective modules in term of supplements, we say that a module M has the property (SE) (respectively, the property (SSE)) if, whenever M ( N, M has a supplement that is a direct summand of N (respectively, a strong supplement in N). We show that a ring R is a left and right artinian serial ring with Rad(R)2 = 0 if and only if every left R-module has the property (SSE). We prove that a commutative ring R is an artinian serial ring if and only if every left R-module has the property (SE).