On $e^*$-Semisimple Modules and the Associated Socle

Abstract

Let $R$ be a ring, and let $M$ be a right $R$-module. Recently, Baannon and Khalid  introduced and studied the concept of $e^*$-essential submodules, extending the notion of essential submodules. In this context, we consider $e^*$-semisimple submodules, and we define the $e^*$-socle of $M$ as the largest $e^*$-semisimple submodule of $M$. It turns out that several properties of the socle can be extended to the $e^*$-socle. It is shown that the $e^*$-socle of $M$ is exactly the intersection of all its $e^*$-essential submodules.