Some properties of graded 2-absorbing and graded weakly 2-absorbing submodules

Abstract

Let G be a group with identity e. Let R be a G-graded commutative ring and M a graded R-module. In this paper we will obtain some results concerning the graded 2-absorbing and graded weakly 2-absorbing submodules of a graded modules over a commutative graded ring.

Graded ϕ-2-Absorbing and Graded ϕ-2-Absorbing Primary Submodules

“…s ∈ K or a ∈ Grad((K : R M))). As graded prime ideals (submodules) have an important role in graded ring (module) theory, several authors generalized these concepts in different ways, see ( [4][5][6][7][8]). Atani in [9] has introduced graded weakly prime submodules.…”

“…Then introducing graded 2-absorbing submodules (resp. graded weakly 2-absorbing submodules) in [5] generalized the concept of graded 2-absorbing ideals (resp. graded weakly 2-absorbing ideals) to graded submodules.…”

On generalizations of graded second submodules

“…The authors in [5] extended graded 2-absorbing ideals to graded 2absorbing submodules. A proper graded R-submodule N of M is said to be graded 2-absorbing if whenever a, b ∈ h(R) and m ∈ h(M ) such that abm ∈ N , then either am ∈ N or bm ∈ N or ab ∈ (N : R M ).…”

“…A proper graded R-submodule N of M is said to be graded 2-absorbing if whenever a, b ∈ h(R) and m ∈ h(M ) such that abm ∈ N , then either am ∈ N or bm ∈ N or ab ∈ (N : R M ). Graded 2-absorbing submodules have been deeply studied in [7].…”

Almost graded multiplication and almost graded comultiplication modules

“…. This concept has been first introduced and studied in [12], and then generalized into graded n-absorbing submodules in [13]. In [11], a parallel study given in [9] has been followed to investigate the new class of graded absorbing multiplication modules, by first providing many interesting results on graded 2-absorbing submodules.…”