Nearly Essential Submodule

Abstract

This paper investigates  the concept of a nearly essential submodule of an-module , together with its fundamental properties and its relationship with essential submodules. The aim of this work is to establish several properties of  nearly essential submodules and to clarify their relationship with essential submodules. In particular, it is shown that each essential submodule is nearly essential, whereas the opposite does not hold overall. Several result concerning the behavior of nearly essential submodules under inclusion, intersection, and sum are established. In particular, if a submodule contains a nearly essential submodule, then it is itself nearly essential. Conditions under which the intersection of  two submodules is nearly essential are also investigated. Moreover, some result concerning essential and nearly essential submodules are obtained in relation to the Jacobson radical  . Furthermore, the relationship between nearly essential submodules and essential submodules in injective modules is studied. Thus, the obtained result demonstrate that the notion of a nearly essential submodule provides a generalization of the concept of an essential submodule