Graded gw-Prime Submodules

Abstract

Let G be a group with identity e, R a commutative G-graded ring with unity 1 and M a G-graded R-module. In this article, graded gw-prime submodules of M are defined. This class of graded submodules is a generalization of graded weakly prime submodules. A graded R-module M is said to be graded gw-prime if whenever L is a graded R-submodule of M and x; y ∊ h(R) such that xyL = {0}, then either x^2L = {0} or y^2L = {0}. A graded R-submodule N of M is said to be graded gw-prime if M/N is a graded gw-prime R-module, i.e., if whenever L is a graded R-submodule of M and x; y ∊ h(R) such that xyL ⊆ N, then either x^2L ⊆ N or y^2L ⊆ N. Also, we introduce the concept of graded valuation modules; let R be a G-graded domain with quotient field F and M a G-graded torsion free R-module. For b = x/y ∊ F where x; y ∊ h(R) and for s ∊ h(M), we write bs ∊ M if there exists t ∊ h(M) such that xs = yt. Then M is said to be a graded valuation R-module if for each b = x/y ∊ K (x; y ∊ h(R)), we have either bM ⊆ M or b^-1 M ⊆ M. After studying general properties of graded gw-prime submodules, their relation with graded valuation modules are examined.