Small - essential sub-modules relative to an arbitrary sub-module

摘要

Let M be an R-module and T a submodule of M. The notion of small-essential and T-essential sub module are well known. In this paper, we introduce small-essential submodule of M relative to T, which is a generalization of both ones. Several properties and characterizations have been introduced. We invest this concept to consider two socles of modules and study some of their properties.