{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,29]],"date-time":"2022-03-29T15:21:45Z","timestamp":1648567305762},"reference-count":7,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":5398,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1999,6]]},"abstract":"<jats:p>Let <jats:italic>CR<\/jats:italic> denote the first-order theory of commutative rings with unity, formulated in the language <jats:italic>L<\/jats:italic> = \u3008 +, \u2022, 0, 1\u3009. Virtually everything that is known about existentially complete (e.c.) models of <jats:italic>CR<\/jats:italic> is contained in Cherlin's paper [2], where it is shown, in particular, that the e.c. models are not first-order axiomatizable. The purpose of this note is to show that, in analogy with the case of fields, there exists a unique prime e.c. model of <jats:italic>CR<\/jats:italic> in each characteristic <jats:italic>n<\/jats:italic> &gt; 2. As a consequence we settle Problem 8 in the list of open questions at the end of Hodges' book <jats:bold><jats:italic>Building models by games<\/jats:italic><\/jats:bold> ([5], p. 278).<\/jats:p><jats:p>By a \u201cprime\u201d e.c. model of characteristic <jats:italic>n<\/jats:italic> \u2265 2 we mean one that embeds in every e.c. model of characteristic <jats:italic>n<\/jats:italic>. (The embedding is not always elementary, since [2] not all e.c. models of characteristic <jats:italic>n<\/jats:italic> are elementarily equivalent.) The prime model is characterized by the fact that it is the union of a chain of finite subrings each of which is an amalgamation base for <jats:italic>CR<\/jats:italic>. In \u00a71 we describe the finite amalgamation bases for <jats:italic>CR<\/jats:italic> and show that every finite model embeds in a finite amalgamation base; in \u00a72 we use this information to obtain prime e.c. models and answer Hodges' question.<\/jats:p><jats:p>Our results on prime e.c. models were obtained some years ago, during the fall term of 1982, while the author was a visitor at Wesleyan University. The author wishes to take this opportunity to thank the mathematics department at Wesleyan for its hospitality during that visit, and subsequent ones.<\/jats:p>","DOI":"10.2307\/2586488","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:03:01Z","timestamp":1146938581000},"page":"629-633","source":"Crossref","is-referenced-by-count":0,"title":["Prime e.c. commutative rings in characteristic <i>n<\/i> \u2265 2"],"prefix":"10.1017","volume":"64","author":[{"given":"Dan","family":"Saracino","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200013505_ref002","first-page":"493\u2013499","volume":"38","author":"Cherlin","year":"1973","journal-title":"Algebraically closed commutative rings"},{"key":"S0022481200013505_ref001","volume-title":"Lecture notes on nilpotent groups","author":"Baumslag","year":"1971"},{"key":"S0022481200013505_ref003","doi-asserted-by":"publisher","DOI":"10.1007\/BF02756563"},{"key":"S0022481200013505_ref004","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0064082"},{"key":"S0022481200013505_ref005","volume-title":"Building models by games","author":"Hodges","year":"1985"},{"key":"S0022481200013505_ref006","volume-title":"Commutative rings","author":"Kaplansky","year":"1974"},{"key":"S0022481200013505_ref007","volume-title":"Finite rings with identity","author":"McDonald","year":"1974"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200013505","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,9]],"date-time":"2019-05-09T21:17:30Z","timestamp":1557436650000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200013505\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1999,6]]},"references-count":7,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1999,6]]}},"alternative-id":["S0022481200013505"],"URL":"https:\/\/doi.org\/10.2307\/2586488","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1999,6]]}}}