some results on graded s-strongly prime submodules

Let g be a group with identity e and r be a commutative g-graded ring with nonzero identity, s ⊆ h(r) a multiplicatively closed subset of r and m be a graded rmodule. a graded submodule n of m with (n :r m)∩s = ∅ is said to be graded s-strongly prime if there exists s ∈ s such that whenever ((n + rxg) :r m)yh ⊆ n, then sxg ∈ n or syh ∈ n for all xg, yh ∈ h(m). the aim of this paper is to introduce and investigate some basic properties of the notion of graded s-strongly prime submodules, especially in graded multiplication modules. moreover, we investigate the behaviour of this structure under graded module homomorphisms, localizations of graded modules, quotient graded modules, cartesianproduct.