On -Quasi-Semiprime Submodules

Abstract

Let G be a group. A ring R is called a graded ring (or G-graded ring) if there exist additive subgroups Rα of R indexed by the elements α∈G such that R=⊕α∈GRαand RαRβ⊆Rαβ for all α, β∈G. If an element of R belongs to h(R)=∪α∈GRα, then it is called homogeneous. A Left R-module M is said to be a graded R-module if there exists a family of additive subgroups {Mα} α∈G of M such that M=⊕α∈GMα and RαMβ⊆Mαβ or all α,β∈G. Also if an element of M belongs to ∪α∈GMα=h(M), then it is called homogeneous. A submodule N of M is said to be a graded submodule of M if N=⊕α∈GN∩Mα:=⊕α∈GNα. Let G be a group with identity e. Let R be a G-graded commutative ring and M a graded R-module. A proper graded submodule S of M is said to be a gr-semiprime submodule if whenever rⁿm∈S where r∈h(R), m∈h(M) and n∈Z⁺, then rm∈S. In this work, we introduce the concept of gr-quasi-semiprime submodule as a generalization of gr-semiprime submodule and give some basic properties of these classes of graded submodules. We say that a proper graded submodule S of M is a gr-quasi-semiprime submodule if (S:RM) is a gr-semiprime ideal of R.

Keywords

• Graded quasi-semiprime submodule
• Graded semiprime submodule
• Graded prime

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