MODULES WITH CHAIN CONDITIONS ON SEMISMALL SUBMODULES

Let R be an associative ring with identity and M be unital non zero R-module. A submodule N of a module M is called a semismall submodule of M (briefly N S M) if N = 0 or for each nonzeror submodule K of N, N / K M / K. In this work,we study this kind of submodule of M and the modules which is satisfies the ascending chain condition (a. c. c.) and descending chain condition (d. c. c.) on semismall submodules .Then we generalize the Rad(M) into s- Rad(M) ,It is equale to the sum of all semismall submodule of M . We show that if N not semismall submodule of M.Then s-Rad (N) = N∩s-Rad (M)and we discuss some of the basic properties of this types of submodules.