Annihilator essential and S-annihilator essential submodules

Let [Formula: see text] be a commutative ring with [Formula: see text] and [Formula: see text] be an [Formula: see text]-module. In this paper, we will present two classes of submodules of [Formula: see text] named annihilator essential and [Formula: see text]-annihilator essential submodules such that [Formula: see text] is a multiplicatively closed subset of [Formula: see text]. A submodule [Formula: see text] of [Formula: see text] is an annihilator essential submodule if [Formula: see text] results that [Formula: see text] for each submodule [Formula: see text] of [Formula: see text]. In addition, [Formula: see text] is an [Formula: see text]-annihilator essential submodule of [Formula: see text] if there exists a fixed element [Formula: see text] such that [Formula: see text] results that [Formula: see text] for each submodule [Formula: see text] of [Formula: see text]. We investigate some results concerning to these classes of submodules analogous with essential submodules and also obtain some basic results.