{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T07:57:09Z","timestamp":1775030229035,"version":"3.50.1"},"reference-count":17,"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":13160,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1978,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We say that a ring <jats:italic>admits elimination of quantifiers<\/jats:italic>, if in the language of rings, {0, 1, +, \u00b7}, the complete theory of <jats:italic>R<\/jats:italic> admits elimination of quantifiers.<\/jats:p><jats:p>Theorem 1. <jats:italic>Let D be a division ring. Then D admits elimination of quantifiers if and only if D is an algebraically closed or finite field<\/jats:italic>.<\/jats:p><jats:p>A ring is <jats:italic>prime<\/jats:italic> if it satisfies the sentence: \u2200<jats:italic>x<\/jats:italic>\u2200<jats:italic>y<\/jats:italic>\u2203<jats:italic>z<\/jats:italic> (<jats:italic>x<\/jats:italic> =0 \u2228 <jats:italic>y<\/jats:italic> = 0\u2228 <jats:italic>xzy<\/jats:italic> \u2260 0).<\/jats:p><jats:p>T<jats:sc>heorem<\/jats:sc> 2. <jats:italic>If R is a prime ring with an infinite center and R admits elimination of quantifiers, then R is an algebraically closed field<\/jats:italic>.<\/jats:p><jats:p>Let <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049926_inline1\"\/> 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=\"S0022481200049926_inline2\"\/> 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=\"S0022481200049926_inline3\"\/> be the class of rings of the form <jats:italic>GF(p<jats:sup>n<\/jats:sup>)\u2295GF(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. Let <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049926_inline4\"\/> be the set of ordered pairs (<jats:italic>f, Q<\/jats:italic>) where <jats:italic>Q<\/jats:italic> is a finite set of primes and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049926_inline5\"\/> such that the characteristic of the ring <jats:italic>f(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=\"S0022481200049926_inline3\"\/> be the class of rings of the form \u2295<jats:sub><jats:italic>q<\/jats:italic> \u2208 <jats:italic>Q<\/jats:italic><\/jats:sub><jats:italic>f(q)<\/jats:italic>, for some (<jats:italic>f, Q<\/jats:italic>) in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049926_inline4\"\/>.<\/jats:p><jats:p>T<jats:sc>heorem<\/jats:sc> 3. <jats:italic>Let R be a finite ring without nonzero trivial ideals. Then R admits elimination of quantifiers if and only if R belongs to<\/jats:italic><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049926_inline3\"\/>.<\/jats:p><jats:p>T<jats:sc>heorem<\/jats:sc> 4. <jats:italic>Let R be a ring with the descending chain condition of left ideals and without nonzero trivial ideals. Then R admits elimination of quantifiers if and only if R is an algebraically closed field or R belongs to<\/jats:italic><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049926_inline3\"\/>.<\/jats:p><jats:p>In contrast to Theorems 2 and 4, we have<\/jats:p><jats:p>T<jats:sc>heorem<\/jats:sc> 5. <jats:italic>If R is an atomless p-ring, then R is finite, commutative, has no nonzero trivial ideals and admits elimination of quantifiers, but is not prime and does not have the descending chain condition<\/jats:italic>.<\/jats:p><jats:p>We also generalize Theorems 1, 2 and 4 to alternative rings.<\/jats:p>","DOI":"10.2307\/2271952","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:46:34Z","timestamp":1146951994000},"page":"92-112","source":"Crossref","is-referenced-by-count":9,"title":["Rings which admit elimination of quantifiers"],"prefix":"10.1017","volume":"43","author":[{"given":"Bruce I.","family":"Rose","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200049926_ref017","first-page":"582","volume-title":"Proceedings of the Royal Academy of Sciences","volume":"57","author":"Tarski","year":"1954"},{"key":"S0022481200049926_ref016","unstructured":"Slater M. , Strongly prime alternative rings (to appear)."},{"key":"S0022481200049926_ref013","doi-asserted-by":"publisher","DOI":"10.1016\/0021-8693(71)90069-X"},{"key":"S0022481200049926_ref012","unstructured":"Rose B. 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