ESSENTIAL SUBMODULES RELATIVE TO A SUBMODULE

In this paper, our aim is to introduce and study the essential submodules of an R-module M relative to an arbitrary submodule T of M. Let T be an arbitrary submodule of an R-module M, then we say that a submodule N of M is an essential submodule of M relative to T, whenever for every submodule X of M, N ∩ X ⊆ T implies that (T : M) ⊆e Ann(X). We investigate some new results concerning to this class of submodules. Among various results we prove that for a faithful multiplication R-module M, if the submodule N of M is an essential submodule of M relative to T, then (N : M) is an essential ideal of R relative to (T : M). The converse is true if M is moreover a finitely generated module.