S. Rajaee - Department of Mathematics, University of Payame Noor,Tehran, Iran.

‎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\cap X\subseteq T implies that‎ ‎(T:M)\subseteq ^{e}{\rm 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.

‎multiplication module, faithful module‎, ‎essential submodule