Some Basic Results on -Graded 2-Absorbing and Graded -Prime Submodules

Abstract

Aim (Background): The concepts of graded prime ideals, graded prime submodules and graded weakly prime submodules have been studied in previous years. Methods: In this paper, we introduce the concepts of  ϕ-graded-2-absorbing, -graded-2-absorbing, graded ϕ-prime and graded -prime submodules, Using above introduced concepts. Results: Let G be a group with identity e, R a G-graded commutative ring, M a graded R-module,  the sets of all submodules of M, Mg , respectively, and  two functions, where . In this paper, we introduce the concepts of -graded-2-absorbing, -graded-2-absorbing, graded -prime and graded -prime submodules and we study some basic properties of them. Also, we give some characterizations of these submodules. Discussion: Prime ideals and prime submodules play a central role in commutative ring theory. In this paper we have tried to extend these concepts.