{"id":3208,"date":"2026-07-17T20:09:51","date_gmt":"2026-07-17T20:09:51","guid":{"rendered":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/?page_id=3208"},"modified":"2026-07-17T20:29:29","modified_gmt":"2026-07-17T20:29:29","slug":"graded-s-r-ideals","status":"publish","type":"page","link":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/graded-s-r-ideals\/","title":{"rendered":"Graded S-r-ideals"},"content":{"rendered":"\n
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Moroccan Journal of Algebra and Geometry with Applications<\/strong><\/p>\n\n\n\n

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Graded S-r-ideals<\/strong><\/p>\n<\/div>\n<\/div>\n\n\n\n

Samir Guesmi<\/strong> <\/i>
Department of Mathematics, Higher School of Sciences and Technologies, Hammam sousse, Tunisia.<\/p>\n\n\n\n

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Pages 60-70 | Received 28 June 2024, Accepted 28 June 2025, Published 19 July 2026 <\/span><\/p>\n\n\n\n

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Abstract<\/span><\/strong><\/p>\n\n\n\n

Let G<\/span> be a group, R<\/span> a G<\/span>-graded commutative ring with nonzero unity 1, and S\\subseteq h(R)<\/span> a multiplicative subset of R<\/span>. This article introduces the concept of graded S\\text{-}r<\/span>-ideal and graded S\\text{-}pr<\/span>-ideal. A proper graded ideal P<\/span> of R<\/span> is termed a graded S\\text{-}r<\/span>-ideal (resp., graded S\\text{-}pr<\/span>-ideal}) if there exists an s\\in S<\/span> such that for any nonunit elements x, y\\in\nh(R)<\/span> such that xy\\in P<\/span> with ann(x)=0<\/span>, then sy\\in P<\/span> (resp., sy\\in Grad(P)<\/span>) with Grad(P)<\/span> being the graded radical of P<\/span>. In this paper, we begin by providing counterexamples illustrating graded S\\text{-}r<\/span>-ideals and graded S\\text{-}pr<\/span>-ideals that do not qualify as graded S<\/span>-prime ideals. Additionally, we present an example of a graded S<\/span>-prime ideal that does not meet the criteria to be considered a graded S\\text{-}r<\/span>-ideal or a graded S\\text{-}pr<\/span>-ideal. Moreover, we investigate some fundamental properties of graded S\\text{-}r<\/span>-ideals (resp., graded S\\text{-}pr<\/span>-ideals), discuss their forms in finite direct products, and study their presence in Nagata’s idealization ring.<\/p>\n\n\n\n

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Keywords<\/strong>:<\/span> Graded S\\text{-}r<\/span>-ideals, Graded S\\text{-}pr<\/span>-ideals, Nagata’s idealization.<\/p>\n\n\n\n

MSC numbers<\/strong>:<\/span> Primary 13A02; Secondary 16W50.<\/p>\n\n\n\n

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Moroccan Journal of Algebra and Geometry with Applications Latest articles 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 be a group, a -graded commutative ring with nonzero unity 1, and Read More<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"template-page-builder.php","meta":{"footnotes":""},"class_list":["post-3208","page","type-page","status-publish","hentry","entry"],"_links":{"self":[{"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages\/3208","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/comments?post=3208"}],"version-history":[{"count":7,"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages\/3208\/revisions"}],"predecessor-version":[{"id":3221,"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages\/3208\/revisions\/3221"}],"wp:attachment":[{"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/media?parent=3208"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}