{"id":2941,"date":"2026-01-03T21:03:55","date_gmt":"2026-01-03T21:03:55","guid":{"rendered":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/?page_id=2941"},"modified":"2026-01-26T12:07:11","modified_gmt":"2026-01-26T12:07:11","slug":"pairs-of-rings-sharing-their-units","status":"publish","type":"page","link":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/pairs-of-rings-sharing-their-units\/","title":{"rendered":"Pairs of rings sharing their units"},"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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Pairs of rings sharing their units<\/strong><\/p>\n<\/div>\n<\/div>\n\n\n\n

Gabriel Picavet<\/strong>\\,^1<\/span><\/i> and Martine Picavet-L’Hermitte\\,^2<\/span>
\\,^{1,2}<\/span> Math\u00e9matiques 8 Rue du Forez, 63670 – Le Cendre France.<\/p>\n\n\n\n


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Pages 207-238 | Received 12 November 2024, Accepted 21 March 2025, Published 26 January 2026 <\/span><\/p>\n\n\n\n

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

We are working in the category of commutative unital rings and denote by \\mathrm U(R)<\/span> the group of units of a nonzero ring R<\/span>. An extension of rings R\\subseteq S<\/span>, satisfying \\mathrm U(R)=R \\cap\\mathrm U(S)<\/span> is usually called local. This paper is devoted to the study of ring extensions such that \\mathrm U(R)=\\mathrm U(S)<\/span>, that we call strongly local. P. M. Cohn in a paper, entitled Rings with zero divisors, introduced some strongly local extensions. We generalized under the name Cohn’s rings his definition and give a comprehensive study of these extensions. As a consequence, we give a constructive proof of his main result. Now Lequain and Doering studied strongly local extensions, where S<\/span> is semilocal, so that S\/\\mathrm J(S)<\/span>, where \\mathrm J(S)<\/span> is the Jacobson radical of S<\/span>, is Von Neumann regular. These rings are usually called J<\/span>-regular. We establish many results on J<\/span>-regular rings in order to get substantial results on strongly local extensions when S<\/span> is J<\/span>-regular. The Picard group of a J<\/span>-regular ring is trivial, allowing to evaluate the group \\mathrm U(S)\/\\mathrm U(R)<\/span> when R<\/span> is J<\/span>-regular. We then are able to give a complete characterization of the Doering-Lequain context. A Section is devoted to examples. In particular, when R<\/span> is a field, the strongly local and weakly strongly inert properties are equivalent.<\/p>\n\n\n\n

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Keywords<\/strong>:<\/span> Group of units, local extension, strongly local extension, J-regular ring, integral extension, FCP extension<\/p>\n\n\n\n

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MSC numbers<\/strong>:<\/span> Primary 13B02; Secondary 13B25.<\/p>\n\n\n\n

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Moroccan Journal of Algebra and Geometry with Applications Latest articles Pairs of rings sharing their units Gabriel Picavet and Martine Picavet-L’Hermitte Math\u00e9matiques 8 Rue du Forez, 63670 – Le Cendre France. Pages 207-238 | Received 12 November 2024, Accepted 21 March 2025, Published 26 January 2026 Abstract We are working in the category of commutative 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-2941","page","type-page","status-publish","hentry","entry"],"_links":{"self":[{"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages\/2941","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=2941"}],"version-history":[{"count":4,"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages\/2941\/revisions"}],"predecessor-version":[{"id":2999,"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages\/2941\/revisions\/2999"}],"wp:attachment":[{"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/media?parent=2941"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}