On Graded 2-n-Submodules of Graded Modules Over Graded Commutative Rings

In this article, all rings are commutative with a nonzero identity. Let G be a group with identity e, R be a G-graded commutative ring, and M be a graded R-module. In 2019, the concept of graded n-ideals was introduced and studied by Al-Zoubi, Al-Turman, and Celikel. A proper graded ideal I of R is said to be a graded n-ideal of R if whenever r, s∈h (R) with rs∈I and r∉Gr (0), then s∈I. In 2023, the notion of graded n-ideals was recently extended to graded n-submodules by Al-Azaizeh and Al-Zoubi. A proper graded submodule N of a graded R-module M is said to be a graded n-submodule if whenever t∈h (R), m∈h (R) with tm∈N and t∉Gr (AnnR (M) ), then m∈N. In this study, we introduce the concept of graded 2-n-submodules of graded modules over graded commutative rings generalizing the concept of graded n-submodules. We investigate some characterizations of graded 2-n-submodules and investigate the behavior of this structure under graded homomorphism and graded localization. A proper graded submodule U of M is said to be a graded 2-n-submodule if whenever r, s∈h (R), m∈ (M) and rsm∈U, then rs∈Gr (AnnR (M) ) or rm∈U or tm∈U.