# E-small essential submodules

Document Type : Research Paper

Authors

1 Department of Physics, College of Education, University of Samarra, Iraq

2 Directorate of Education Salah Eddin, Khaled Ibn Al Walid School, Tikrit, Iraq

Abstract

Let $R$ be a commutative ring with identity, and $U_{R}$ be an $R$-module, with $E = End(U_{R})$. In this work we consider a generalization of class small essential submodules namely E-small essential submodules. Where the submodule $Q$ of $U_{R}$ is said E-small essential if $Q$ $\cap W = 0$ , when W is a small submodule of $U_{R}$, implies that $N_{S}\left( W \right) = 0$, where $N_{S}\left( W \right) = \left\{ \psi \in E\ |\ Im\psi \subseteq W \right\}$. The intersection ${\overline{B}}_{R}(U)$ of each submodule of $U_{R}$ contained in $Soc(U_{R})$. The ${\overline{B}}_{R}(U)$ is unique largest E-small essential submodule of $U_{R}$, if $U_{R}$ is cyclic. Also in this paper we study ${\overline{B}}_{R}(U)$ and ${\overline{W}}_{E}\left( U \right)$. The condition when ${\overline{B}}_{R}(U)$ is E-small essential, and $\text{Tot}\left( \ U,U \right) = {\overline{W}}_{E}\left( U \right) = J(E)$ are given.

Keywords