graded i-prime submodules

Abstract. let r = ⊕ g2g rg be a ggraded commutative ring with identity, i be a graded ideal and let m be a ggraded unitary r-module, where g is a semigroup with identity e. we introduce graded iprime 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 iprime if and only if f p is graded iprime submodule of f m. as an application, for fin