S-Generalized supplemented modules

الملخص EN

Xue introduced the following concept : Let M be an R- module.

M is called a generalized supplemented module if for every sub module N of M, there exists a sub module K of M such that M = N +K and N∩K Rad (K) N.

Hamada and B.

AL- Hashimi introduced the following concept : Let S be a property on modules.

S is called a quasi - radical property if the following conditions are satisfied : 1.

For every epimorphism f : M → N, where M and N are any two R- modules.

If the module M has the property S, then the module N has the property S.

2.

Every module M contained the sub module S (M).

These observations lead us to introduce S- generalized supplemented modules.

Let S be a quasi- radical property.

We say that an R-module M is S- generalized supplemented module if for every sub module N of M, there exists a sub module K of M such that M = N + K and N K S(K).

The main purpose of this work is to develop the properties of S-generalized supplemented modules.

Many interesting and useful results are obtained about this concept.

We illustrate the concepts, by examples.