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S-Generalized supplemented modules | ||
| Baghdad Science Journal | ||
| Article 1, Volume 7, Issue 0, 2010, Pages 180-190 | ||
| Authors | ||
| A. J. Al-Rikabiy; B. H. Al-Bahrany | ||
| Abstract | ||
| 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, then the module N 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. | ||
| Keywords | ||
| quasi; radical property; generalized supplemented module; small submodule | ||
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