Document Type : Research Paper

Authors

• Mamoon F. Khalf ¹
• Hind Fadhil Abbas ²

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 $N_S\left(W\right)=\{\psi \in E|Im\psi \subseteq W\}$. The intersection ${\bar{B}}_R(U)$ of each submodule of $U_R$ contained in $Soc(U_R)$. The ${\bar{B}}_R(U)$ is unique largest E-small essential submodule of $U_R$, if $U_R$ is cyclic. Also in this paper we study ${\bar{B}}_R(U)$ and ${\bar{W}}_E\left(U\right)$. The condition when ${\bar{B}}_R(U)$ is E-small essential, and $\text{Tot}(U,U)={\bar{W}}_E\left(U\right)=J(E)$ are given.

Keywords

• Small submodule
• Small essential submodules
• E-small essential submodules
• Endomorphism ring