Abstract
Abstract Let G be a group with identity e, R be a G-graded commutative ring with a nonzero unity 1, I be a graded ideal of R, and M be a G-graded R-module. In this article, we introduce the concept of graded I-second submodules of M as a generalization of graded second submodules of M and achieve some relevant outcomes.
Highlights
A proper graded ideal P of R is said to be graded prime if whenever x, y ∈ h(R) such that xy ∈ P, either x ∈ P or y ∈ P
Graded prime submodules have been introduced by Atani in [2]
A proper graded R-submodule N of M is said to be graded prime if whenever r ∈ h(R) and m ∈ h(M) such that rm ∈ N, either m ∈ N or r ∈ (N :R M)
Summary
A proper graded ideal P of R is said to be graded prime if whenever x, y ∈ h(R) such that xy ∈ P, either x ∈ P or y ∈ P. A nonzero graded R-submodule N of M is said to be graded second if for each a ∈ h(R), the graded R-homomorphism f : N → N defined by f (x) = ax is either surjective or zero. In this case, AnnR(N) is a graded prime ideal of R. The main purpose of this article is to follow [13] in order to introduce and study the concept of graded I-second submodules of a graded R-module M as a generalization of graded second submodules of M and achieve some relevant outcomes. We follow [14] to introduce the concept of graded I-prime ideals of a graded ring R, we show that a graded prime ideal is a graded I-prime ideal for every graded ideal I of R, but
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