{"id":2251,"date":"2024-06-29T10:35:21","date_gmt":"2024-06-29T10:35:21","guid":{"rendered":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/?page_id=2251"},"modified":"2024-07-02T21:02:29","modified_gmt":"2024-07-02T21:02:29","slug":"%cf%86-rings-from-a-module-theoretic-point-of-view-a-survey","status":"publish","type":"page","link":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/%cf%86-rings-from-a-module-theoretic-point-of-view-a-survey\/","title":{"rendered":"\u03c6-rings from a module-theoretic point of view: a survey"},"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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\\phi<\/span>-rings from a module-theoretic point of view: a survey<\/strong><\/p>\n<\/div>\n<\/div>\n\n\n\n

Hwankoo Kim\\,^1<\/span>, Najib Mahdou<\/strong>\\,^2<\/span> <\/i>, and El Houssaine Oubouhou\\,^2<\/span>
^1\\,<\/span>Division of Computer Engineering, Hoseo University, Asan 31499, Republic of Korea.
^2\\,<\/span>Laboratory of Modelling and Mathematical Structures,
Department of Mathematics, Faculty of Science and Technology of Fez, Box 2202, University S.M. Ben Abdellah Fez, Morocco.<\/p>\n\n\n\n

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Pages 78-114 | Received 02 September<\/span> 2023, Accepted 13 October 2023, Published 30 June 2024 <\/span><\/p>\n\n\n\n

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

Let R<\/span> be a commutative ring with a nonzero identity and Nil(R)<\/span> be its set of nilpotent elements. Recall that a prime ideal P<\/span> of R<\/span> is called divided prime if P \\subset (x)<\/span> for every x \\in R\\setminus P<\/span>; thus a divided prime ideal is comparable to every ideal of R<\/span>. In many articles, the authors investigated the class of rings \\mathcal{H} = \\{R\\ |\\ R \\text{ is a commutative ring and } Nil(R) \\text{ is a\ndivided prime ideal of } R\\}<\/span> (observe that if R<\/span> is an integral domain, then R \\in \\mathcal{H}<\/span>). If R \\in \\mathcal{H}<\/span>, then R<\/span> is called a \\phi<\/span>-ring. In this paper, we survey known results concerning \\phi<\/span>-rings from a module-theoretic point of view.<\/p>\n\n\n\n

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Keywords<\/strong>:<\/span>  \\phi<\/span>-ring, \\phi<\/span>-Pr\u00fcfer, nonnil-Noetherian, \\phi<\/span>-Dedekind, \\phi<\/span>-torsion, \\phi<\/span>-flat, nonnil-injective, \\phi<\/span>-von Neumann regular, nonnil-coherent, nonnil-commutative diagram, \\phi<\/span>-(weakly) global dimension, nonnil-projective.<\/p>\n\n\n\n

MSC numbers<\/strong>:<\/span> Primary 13A15, 13C11, 13D05; Secondary 13A15, 13B02, 3C05, 13C12, 13E15, 13F05.<\/p>\n\n\n\n

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Moroccan Journal of Algebra and Geometry with Applications Latest articles -rings from a module-theoretic point of view: a survey Hwankoo Kim, Najib Mahdou , and El Houssaine Oubouhou Division of Computer Engineering, Hoseo University, Asan 31499, Republic of Korea.Laboratory of Modelling and Mathematical Structures,Department of Mathematics, Faculty of Science and Technology of Fez, Box 2202, 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-2251","page","type-page","status-publish","hentry","entry"],"_links":{"self":[{"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages\/2251","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=2251"}],"version-history":[{"count":6,"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages\/2251\/revisions"}],"predecessor-version":[{"id":2331,"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages\/2251\/revisions\/2331"}],"wp:attachment":[{"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/media?parent=2251"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}