{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,5]],"date-time":"2022-04-05T23:08:27Z","timestamp":1649200107012},"reference-count":2,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":12795,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1979,3]]},"abstract":"<jats:p>Matatyahu Rubin pointed out that the proof of Lemma 6.1 [2] works only for rings of prime or zero characteristic. This invalidates the characterization of semiprime rings with the descending chain condition on right or left ideals which admit elimination of quantifiers given in [2] and cited in the abstract [1]. Although the correct characterization is easy to derive, it is complex to state.<\/jats:p><jats:p>Let <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048775_inline10\" \/> be the class of finite fields. Let <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048775_inline20\" \/> be the class of 2 \u00d7 2 matrix rings over a field with a prime number of elements. Let <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048775_inline30\" \/> be the class of rings of the form <jats:italic>GF<\/jats:italic>(<jats:italic>p<jats:sup>n<\/jats:sup><\/jats:italic>) \u2295 <jats:italic>GF<\/jats:italic>(<jats:italic>p<jats:sup>k<\/jats:sup><\/jats:italic>) such that either <jats:italic>n<\/jats:italic> = <jats:italic>k<\/jats:italic> or g.c.d.(<jats:italic>n, k<\/jats:italic>) = 1 and <jats:italic>p<\/jats:italic> is a prime. Let <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048775_inline50\" \/> \u2032 be the class of algebraically closed fields. Let <jats:italic>P<\/jats:italic> denote the set of all prime numbers together with zero. Let <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048775_inline50\" \/>be the set of all ordered pairs (<jats:italic>f, Q<\/jats:italic>) where <jats:italic>Q<\/jats:italic> is a finite subset of <jats:italic>P<\/jats:italic> and <jats:italic>f<\/jats:italic>: <jats:italic>Q<\/jats:italic> \u2192 <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048775_inline10\" \/> \u22c3 <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048775_inline20\" \/> \u22c3 <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048775_inline30\" \/> \u22c3 <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048775_inline40\" \/> such that the characteristic of the ring <jats:italic>f<\/jats:italic>(<jats:italic>q<\/jats:italic>) is <jats:italic>q<\/jats:italic>. Finally, let <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048775_inline60\" \/> be the class of rings of the form \u2295<jats:sup><jats:italic>q<\/jats:italic>\u2208Q<\/jats:sup><jats:italic>f<\/jats:italic>(<jats:italic>q<\/jats:italic>) for some (<jats:italic>f<\/jats:italic>,<jats:italic>Q<\/jats:italic>) in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048775_inline50\" \/>.<\/jats:p><jats:p>A corrected version of Theorem 6.2 [2] is<\/jats:p><jats:p>Theorem 1. <jats:italic>Let R be a ring with the descending chain condition on left or right ideals and without nonzero trivial ideals. Then R admits elimination of quantifiers if only if R belong to<\/jats:italic><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200048775_inline60\" \/>.<\/jats:p>","DOI":"10.2307\/2273709","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:49:34Z","timestamp":1146937774000},"page":"109-110","source":"Crossref","is-referenced-by-count":0,"title":["Corrigendum: \u201cRings which admit elimination of quantifiers\u201d"],"prefix":"10.1017","volume":"44","author":[{"given":"Bruce I.","family":"Rose","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200048775_ref001","first-page":"366","volume":"43","author":"Rose","year":"1978","journal-title":"Rings which admit elimination of quantifiers"},{"key":"S0022481200048775_ref002","first-page":"92","volume":"43","author":"Rose","year":"1978","journal-title":"Rings which admit elimination of quantifiers"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200048775","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,26]],"date-time":"2019-05-26T17:57:01Z","timestamp":1558893421000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200048775\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1979,3]]},"references-count":2,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1979,3]]}},"alternative-id":["S0022481200048775"],"URL":"https:\/\/doi.org\/10.2307\/2273709","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1979,3]]}}}