Any right R-module M is called a CS-module if every submodule of M is
essential in a direct summand of M. A ring is said to be CS-ring if R as a right
R-module is CS [9]. In this paper we study semiperfect ring in which each
simple right R-module
is essential in a direct summand of R. We call such ring
as a extending for simple R-module. Here we find that for such rings, every
simple R-module is weakly-injective if and only if R is weakly-injective if and
only if R is self-injective if and only if R is weakly-semisimple. Examples are
constructed for which simple R-module is essential in a direct summand.
