Modules with chain conditions on semismall submodules

Abstract EN

Let R be an associative ring with identity and M be unital non zero R-module.

A sub module N of a module M is called a semi small sub module of M (briefly N << S M) if N = 0 or for each nonzeror sub module K of N, N / K << M / K.

In this work, we study this kind of sub module of M and the modules which is satisfies the ascending chain condition (a.

c.

c.) and descending chain condition (d.

c.

c.) on semi small sub modules Then we generalize the Rad (M) into s- Rad (M), It is equale to the sum of all semi small sub module of M.

We show that if N not semi small sub module of M.

Then s-Rad (N = N ∩ s-Rad (M) and we discuss some of the basic properties of this types of sub modules.