{"id":1389,"date":"2022-11-15T13:54:22","date_gmt":"2022-11-15T13:54:22","guid":{"rendered":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/?page_id=1389"},"modified":"2022-12-05T22:18:41","modified_gmt":"2022-12-05T22:18:41","slug":"graded-mn-closed-and-graded-weakly-mn-closed-ideals","status":"publish","type":"page","link":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/graded-mn-closed-and-graded-weakly-mn-closed-ideals\/","title":{"rendered":"Graded (m,n)-closed and graded weakly (m,n)-closed 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 (m,n)-closed and graded weakly (m,n)-closed ideals<\/strong><\/em><\/p>\n<\/div>\n<\/div>\n\n\n\n

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Malik Tusif Ahmed <\/strong><\/i>, Mohammed Issoual and Najib Mahdou
Department of Mathematics and Statistics, Hazara University, Mansehra, KPK, Pakistan
CRMEF Khmisset, Morocco
Faculty of Science and Technology of Fez, University S.M. Ben Abdellah Fez, Morocco<\/p>\n\n\n\n

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Pages 392-401 | Received 29 May 2022, Accepted 06 September 2022, Published 06 December 2022 <\/span><\/p>\n\n\n\n

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

Let G<\/span> be a group with identity e<\/span> and R<\/span> a G<\/span>-graded commutative ring with 1\\neq 0<\/span>. In this paper we introduce the
concept of the graded (m,n)<\/span>-closed ideals and graded weakly (m,n)<\/span>-closed ideals. A graded proper ideal I<\/span> of R<\/span> is called a graded (m,n)<\/span>-closed (resp. graded weakly (m,n)<\/span>-closed) ideal if whenever a^m \\in I<\/span> (resp., 0\\neq a^m \\in I<\/span>) for a \\in h(R)<\/span>, then a^n \\in I<\/span>. Many results are given, in particular we investigate the graded (weakly) (m,n)<\/span>-closed ideals in the direct product R_1 \\times R_2<\/span> of G<\/span>-graded rings R_1,\\ R_2<\/span> and in the trivial extension R(+)M<\/span> of a G<\/span>-graded ring R<\/span> by a graded R<\/span>-module M<\/span>.<\/p>\n\n\n\n

Keywords<\/strong>:<\/span>  G<\/span>-graded rings; Trivial extensions, (m,n)<\/span>-closed ideals; Graded (m,n)<\/span>-closed ideals; weakly (m,n)<\/span>-closed ideals; Graded weakly (m,n)<\/span>-closed ideals. <\/p>\n\n\n\n

MSC numbers<\/strong>:<\/span> Primary 13A02, Secondary 13A15, 47B47.<\/p>\n\n\n\n

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0102MJAGA392_401<\/a>\n

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Moroccan Journal of Algebra and Geometry with Applications Latest articles Graded (m,n)-closed and graded weakly (m,n)-closed ideals Malik Tusif Ahmed , Mohammed Issoual and Najib Mahdou Department of Mathematics and Statistics, Hazara University, Mansehra, KPK, Pakistan CRMEF Khmisset, Morocco Faculty of Science and Technology of Fez, University S.M. Ben Abdellah Fez, Morocco Pages 392-401 | Received 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-1389","page","type-page","status-publish","hentry","entry"],"_links":{"self":[{"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages\/1389","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=1389"}],"version-history":[{"count":17,"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages\/1389\/revisions"}],"predecessor-version":[{"id":1563,"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages\/1389\/revisions\/1563"}],"wp:attachment":[{"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/media?parent=1389"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}