Graded I-Prime Submodules

Let R= ₆ ₆ Rg be a G-graded commutative ring with identity, I be a graded ideal and let M a G-graded unitary R-module, where G is a semigroup with identity e. We introduce graded I-prime ideals (submodules) as a generalizations of the classical notions of prime ideals (submodules). We show that the new notions inherite the basic properties of the classical ones. In particular, we investigate the localization theory of these two concepts. We prove that for a faithfull flat module F, a graded submodule P of M is I-prime if and only if F P is graded I-prime submodule of F M. As an application, for finitely generated graded module M over Noetherian graded ring R, the completion of graded I-prime submodules is I-prime submodule.