Z-prime gamma submodule of gamma modules

Document Type : Research Paper


1 Iraqi Ministry of Education, General Directorate of Education for the Holy Karbala, Karbala, Iraq.

2 College of Engineering Al-musayab, Department of Energy Engineering, University of Babylon, Babil, Iraq

3 Information Technology College, Department of Software, University of Babylon, Iraq.


Let $R$ be a $\Gamma$-ring and $\partial$ be an $R\Gamma$-module. A proper $R\Gamma$-submodule. $T$ of an $R\Gamma$-module $\partial$ is called Z-prime $R\Gamma$-submodule if for each $t\in \partial, \gamma \in \Gamma$ and $f \in \partial^{\ast}=Hom_{R_{\Gamma}}(\partial,R),f(t)\gamma t \in T$ implies that either $t \in T$ or $f(t)\in [T: _{R_{\Gamma}} \ \partial]$. The purpose of this paper is to introduce interesting theorems and properties of Z- prime $R\Gamma$-submodule of $R\Gamma$-module and the relation of Z-prime $R\Gamma$-submodule, which represents of generalization Z-prime R-submodule of R-module.