S-Generalized supplemented modules

Xue introduced the following concept: Let M be an R- module. M is called a generalized supplemented module if for every submodule N of M, there exists a submodule 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, 2.Every module M contained the submodule 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 submodule N of M, there exists a submodule 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.