E-small essential submodules

Document Type : Research Paper


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

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


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.