Prime Submodules of Graded Modules

Let G be a group, R be a G-graded ring and M be a G-graded R-module. Suppose P is a prime ideal of Re and g G G. In this article, we define Mg (P) = {m G Mg : Am C PMg for some ideal A of Re satisfying A C P}that is an Re-submodule of Mg, and we investigate some results on this submodule. Also, we introduce a situation where if N is a gr-prime R-submodule of M ,then (Ng : Mg) is a maximal ideal of Re. We close this article by introducing a situation where if N is a gr- R-submodule ofM such that Ne is a weakly prime Re-submodule of Me, then Ng is a prime Re-submodule of Mg.