On gr-n-submodules of graded modules over graded commutative rings

Let G be a group with identity e. Let ℜ be a G-graded commutative ring, ℑ be a graded ℜ -module. In this paper, we introduce and study the concept of graded n-submodules of ℑ. We obtain many results concerning graded n-submodules. Some characterizations of graded n-submodules and their homogeneous components are given. A proper graded submodule U of ℑ is said to be a graded n-submodule if whenever r ∈ h(ℜ), m ∈ h(ℑ) with rm ∈ U and r ∉ Gr(Annℜ(ℑ)), then m ∈ U.