MODULES WITH CHAIN CONDITIONS ON SEMISMALL SUBMODULES

Abstract

Let R be an associative ring with identity and M be unital non zero R-module. Asubmodule 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 thiskind of submodule of M and the modules which is satisfies the ascending chaincondition (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 semismallsubmodule 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.