essential submodules relative to asubmodule

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.