{"id":1991,"date":"2023-12-08T15:53:11","date_gmt":"2023-12-08T15:53:11","guid":{"rendered":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/?page_id=1991"},"modified":"2023-12-11T13:34:09","modified_gmt":"2023-12-11T13:34:09","slug":"direct-limits-and-minimal-ring-extensions","status":"publish","type":"page","link":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/direct-limits-and-minimal-ring-extensions\/","title":{"rendered":"Direct limits and minimal ring extensions"},"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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Direct limits and minimal ring extensions<\/strong><\/p>\n<\/div>\n<\/div>\n\n\n\n

David E. Dobbs <\/strong><\/i>
Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1320, USA.<\/p>\n\n\n\n

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Pages 226\u2013245 | Received 04 March 2023, Accepted 08 August 2023, Published 11 December 2023 <\/span><\/p>\n\n\n\n

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

If \\{A_i \\rightarrow B_i\\}<\/span> is a directed system of minimal (unital) ring extensions (involving associative unital rings that need not be commutative) and the canonical injection A:=\\varinjlim_i A_i \\to B:= \\varinjlim_i B_i<\/span> is used to view A<\/span> as a subring of B<\/span>, then either A=B<\/span> or A \\subset B<\/span> is a minimal ring extension. The preceding assertion is the case n=1<\/span> of a more general result which assumes that there exists an integer n\\geq 0<\/span> such that for each i<\/span>, each chain of rings contained between A_i<\/span> and B_i<\/span> has length at most n<\/span>. For commutative rings, an (upward-)directed union of ramified (resp., decomposed) minimal ring extensions A_i \\rightarrow B_i<\/span> for which each (A_i,M_i)<\/span> is quasi-local, M_j \\cap B_i=M_i<\/span> whenever i \\leq j<\/span> in I<\/span>, and each transition map A_i \\rightarrow A_j<\/span> is an integral extension produces a minimal ring extension A:=\\varinjlim_i A_i \\to \\varinjlim_i B_i=:B<\/span> (that is, \\cup_i A_i \\rightarrow \\cup_i B_i<\/span>) such that if M:=\\cup_i M_i<\/span>, then the minimal ring extension A\/M \\subset B\/M<\/span> is ramified (resp., decomposed) and A \\subset B<\/span> is a minimal ring extension. Applications involving denumerable (upward-)directed unions of fields whose “steps” are algebraic are given to algebraically closed fields and to perfect closures (in the sense of Bourbaki), by using the \\mu<\/span>-field extensions of Gilbert and Quigley.<\/p>\n\n\n\n

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Keywords<\/strong>:<\/span>  Unital associative ring, minimal ring extension, field, noncommutative, field extension, \u03bb-field extension,\u00b5-field extension, algebraically closed field, perfect field, perfect closure, direct limit, length of a chain. <\/p>\n\n\n\n

MSC numbers<\/strong>:<\/span> Primary 16B99; Secondary 12F05, 12F15, 13B99.<\/p>\n\n\n\n

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Moroccan Journal of Algebra and Geometry with Applications Latest articles Direct limits and minimal ring extensions David E. Dobbs  Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1320, USA. Pages 226\u2013245 | Received 04 March 2023, Accepted 08 August 2023, Published 11 December 2023 Abstract If is a directed system of minimal (unital) ring extensions 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-1991","page","type-page","status-publish","hentry","entry"],"_links":{"self":[{"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages\/1991","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=1991"}],"version-history":[{"count":9,"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages\/1991\/revisions"}],"predecessor-version":[{"id":2083,"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages\/1991\/revisions\/2083"}],"wp:attachment":[{"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/media?parent=1991"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}