{"id":2017,"date":"2023-12-08T19:22:04","date_gmt":"2023-12-08T19:22:04","guid":{"rendered":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/?page_id=2017"},"modified":"2023-12-11T13:38:15","modified_gmt":"2023-12-11T13:38:15","slug":"on-the-nature-and-number-of-the-noncommutative-minimal-ring-extensions-of-a-field","status":"publish","type":"page","link":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/on-the-nature-and-number-of-the-noncommutative-minimal-ring-extensions-of-a-field\/","title":{"rendered":"On the nature and number of the noncommutative minimal ring extensions of a field"},"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 the nature and number of the noncommutative minimal ring
extensions of a field <\/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<\/p>\n\n\n\n

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Pages 272\u2013290 | Received 06 June 2023, Accepted 05 September 2023, Published 11 December 2023 <\/span> <\/span><\/p>\n\n\n\n

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

It is known that if k is a field, then any noncommutative minimal ring extension of k is either a prime ring or a non-semiprime ring. Our results include the following. If R is a ring, the collection of R-isomorphism classes represented by minimal ring extensions of R is a set. If X is an indeterminate over a field k, the cardinal number of the set of k(X)-isomorphism classes represented by noncommutative minimal ring extensions of k(X) is infinite, at least denumerably many of those classes have representatives which are simple (and left- and right-Artinian) rings (and, hence, prime rings), and if one also assumes that the field k is infinite, then at least denumerably many of those classes have representatives which are non-semiprime rings. If k is any algebraically closed field (more generally, a field with infinitely many automorphisms), the set of k-isomorphism classes represented by noncommutative minimal ring extensions of k is infinite and the representatives of infinitely many of those classes are non-semiprime rings. If k is a finite field of characteristic p and of cardinality pn with n > 1, the cardinal number of the set of k-isomorphism classes represented by noncommutative minimal ring extensions of k is at least n, and at least n\u22121 of those classes can be represented by a non-semiprime ring. If p is a prime number, then for finite fields k of characteristic p, there is no absolute finite upper bound on the cardinal number of the set of k-isomorphism classes represented by noncommutative minimal ring extensions of k that are simple (hence, prime) rings.<\/p>\n\n\n\n

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Keywords<\/strong>:<\/span>  Unital associative ring, minimal ring extension, field, simple ring, prime ring, noncommutative, field extension, finite field, algebraically closed field, Dorroh extension, rng, bimodule, L\u00fcroth\u2019s Theorem, skew polynomial ring. <\/p>\n\n\n\n

MSC numbers<\/strong>:<\/span> Primary 16B99; Secondary 12F05, 12F10, 16S50, 16D20, 13B99, 12F20, 16P10.<\/p>\n\n\n\n

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MJAGA_Vol2_Issue2-272-290<\/a>\n

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Moroccan Journal of Algebra and Geometry with Applications Latest articles On the nature and number of the noncommutative minimal ring extensions of a field David E. Dobbs Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1320 Pages 272\u2013290 | Received 06 June 2023, Accepted 05 September 2023, Published 11 December 2023 Abstract It is known 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-2017","page","type-page","status-publish","hentry","entry"],"_links":{"self":[{"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages\/2017","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=2017"}],"version-history":[{"count":7,"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages\/2017\/revisions"}],"predecessor-version":[{"id":2086,"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages\/2017\/revisions\/2086"}],"wp:attachment":[{"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/media?parent=2017"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}