Graded S-r-ideals
Samir Guesmi
Department of Mathematics, Higher School of Sciences and Technologies, Hammam sousse, Tunisia.
Pages 60-70 | Received 28 June 2024, Accepted 28 June 2025, Published 19 July 2026
Abstract
Let G be a group, R a G-graded commutative ring with nonzero unity 1, and S\subseteq h(R) a multiplicative subset of R. This article introduces the concept of graded S\text{-}r-ideal and graded S\text{-}pr-ideal. A proper graded ideal P of R is termed a graded S\text{-}r-ideal (resp., graded S\text{-}pr-ideal}) if there exists an s\in S such that for any nonunit elements x, y\in h(R) such that xy\in P with ann(x)=0, then sy\in P (resp., sy\in Grad(P)) with Grad(P) being the graded radical of P. In this paper, we begin by providing counterexamples illustrating graded S\text{-}r-ideals and graded S\text{-}pr-ideals that do not qualify as graded S-prime ideals. Additionally, we present an example of a graded S-prime ideal that does not meet the criteria to be considered a graded S\text{-}r-ideal or a graded S\text{-}pr-ideal. Moreover, we investigate some fundamental properties of graded S\text{-}r-ideals (resp., graded S\text{-}pr-ideals), discuss their forms in finite direct products, and study their presence in Nagata’s idealization ring.
Keywords: Graded S\text{-}r-ideals, Graded S\text{-}pr-ideals, Nagata’s idealization.
MSC numbers: Primary 13A02; Secondary 16W50.
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