{"id":919,"date":"2022-03-29T12:41:23","date_gmt":"2022-03-29T12:41:23","guid":{"rendered":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/?page_id=919"},"modified":"2022-07-04T11:14:07","modified_gmt":"2022-07-04T11:14:07","slug":"on-rings-whose-finitely-generated-flat-submodules-of-free-modulesare-finitely-presented","status":"publish","type":"page","link":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/on-rings-whose-finitely-generated-flat-submodules-of-free-modulesare-finitely-presented\/","title":{"rendered":"On rings whose finitely generated flat submodules of free modules
are finitely presented"},"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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On rings whose finitely generated flat submodules of free<\/strong><\/em><\/em>
modules are finitely presented<\/strong><\/em><\/em><\/p>\n<\/div>\n<\/div>\n\n\n\n

Samir Bouchiba <\/strong><\/i>, Mouhssine El-Arabi & Youssef Najem
Department of Mathematics, Faculty of sciences, Moulay Ismail university of Meknes, Meknes, Morocco<\/p>\n\n\n\n

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Pages 122-131 | Received 15 October<\/span><\/span> 2021, Accepted 02 February 2022, Published 01 June<\/span><\/span><\/span> 2022 <\/span><\/p>\n\n\n\n

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

A ring R<\/span> is right coherent if any finitely generated submodule of a free right module is finitely presented. The main goal of this paper is to study the rings on which any finitely generated flat submodule of a free right module is finitely presented, and thus projective. For this sake, we investigate homological properties of specific classes of modules, namely, the \\mathcal F_1<\/span>-flat modules and the \\mathcal F_1^{\\mathrm{fp}}<\/span>-flat modules, where \\mathcal F_1<\/span> stands for the class of right modules of flat dimension at most one and \\mathcal F_1^{\\mathrm{fp}}<\/span> its subclass consisting of finitely presented elements. The introduced class of rings that we term \\mathcal F_1^{\\mathrm{fp}}<\/span>-coherent rings turns out to behave nicely with respect to \\mathcal F_1^{\\mathrm{fp}}<\/span>-flat modules as do coherent rings with respect to flat modules. In fact, we prove that a ring R<\/span> is \\mathcal F_1^{\\mathrm{fp}}<\/span>-coherent if and only if any direct product of \\mathcal F_1^{\\mathrm{fp}}<\/span>-flat modules is \\mathcal F_1^{\\mathrm{fp}}<\/span>-flat. Also, we show that the class of \\mathcal F_1^{\\mathrm{fp}}<\/span>-coherent rings includes coherent rings, perfect rings, semi-hereditary rings and integral domains.<\/p>\n\n\n\n

Keywords<\/strong>:<\/span>  \\mathcal F_1<\/span>-flat module, \\mathcal F_1^{\\mathrm{fp}}<\/span>-flat module, \\mathcal F_1^{\\mathrm{fp}}<\/span>-coherent ring, finitistic flat dimension, little finitistic dimension. <\/p>\n\n\n\n

MSC numbers<\/strong>:<\/span> Primary 13D02, 13D05, 13D07; Secondary 16E05, 16E10. <\/p>\n\n\n\n

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Moroccan Journal of Algebra and Geometry with Applications Latest articles On rings whose finitely generated flat submodules of free modules are finitely presented Samir Bouchiba , Mouhssine El-Arabi & Youssef Najem Department of Mathematics, Faculty of sciences, Moulay Ismail university of Meknes, Meknes, Morocco Pages 122-131 | Received 15 October 2021, Accepted 02 February 2022, 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-919","page","type-page","status-publish","hentry","entry"],"_links":{"self":[{"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages\/919","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=919"}],"version-history":[{"count":10,"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages\/919\/revisions"}],"predecessor-version":[{"id":1120,"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages\/919\/revisions\/1120"}],"wp:attachment":[{"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/media?parent=919"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}