{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,18]],"date-time":"2026-06-18T13:31:39Z","timestamp":1781789499096,"version":"3.54.5"},"reference-count":12,"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":14164,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1975,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>It is shown that the first-order theory Th<jats:sub><jats:italic>R<\/jats:italic><\/jats:sub>(<jats:italic>A<\/jats:italic>) of a countable module over an arbitrary countable ring <jats:italic>R<\/jats:italic> is <jats:bold>\u2135<\/jats:bold><jats:sub>0<\/jats:sub>-categorical if and only if <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200053767_inline1\"\/><jats:italic>A<jats:sub>i<\/jats:sub><\/jats:italic> finite, <jats:italic>n<\/jats:italic> \u2208 <jats:italic>\u03c9<\/jats:italic>, <jats:italic>\u03ba<jats:sub>i<\/jats:sub><\/jats:italic> \u2264 <jats:italic>\u03c9<\/jats:italic>. Furthermore, Th<jats:italic><jats:sub>R<\/jats:sub>(A)<\/jats:italic> is <jats:bold>\u2135<\/jats:bold><jats:sub>0<\/jats:sub>-categorical for all <jats:italic>R<\/jats:italic>-modules <jats:italic>A<\/jats:italic> if and only if <jats:italic>R<\/jats:italic> is finite and there exist only finitely many isomorphism classes of indecomposable <jats:italic>R<\/jats:italic>-modules.<\/jats:p>","DOI":"10.2307\/2271901","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:35:13Z","timestamp":1146936913000},"page":"213-220","source":"Crossref","is-referenced-by-count":18,"title":["\u2135<sub>0<\/sub>-categorical modules"],"prefix":"10.1017","volume":"40","author":[{"given":"Walter","family":"Baur","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200053767_ref011","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1969.28.699"},{"key":"S0022481200053767_ref010","doi-asserted-by":"publisher","DOI":"10.4064\/fm-41-2-203-271"},{"key":"S0022481200053767_ref008","volume-title":"Saturated model theory","author":"Sacks","year":"1972"},{"key":"S0022481200053767_ref007","first-page":"909","article-title":"Aspects logique de la puret\u00e9 dans les modules","volume":"271","author":"Sabbagh","year":"1970","journal-title":"Comptes Rendus Hebdomadaires des S\u00e9ances de l'Academie des Sciences"},{"key":"S0022481200053767_ref005","doi-asserted-by":"publisher","DOI":"10.4064\/fm-47-1-57-103"},{"key":"S0022481200053767_ref004","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(71)90016-7"},{"key":"S0022481200053767_ref002","volume-title":"Stability of non-model-complete theories, products, groups","author":"Berthier","year":"1973"},{"key":"S0022481200053767_ref001","volume-title":"Introduction to commutative algebra","author":"Atiyah","year":"1969"},{"key":"S0022481200053767_ref012","first-page":"195","article-title":"On \u03c90-categoricity of powers","volume":"17","author":"Waszkiewicz","year":"1969","journal-title":"Bulletin de l'Academie Polonaise des Sciences"},{"key":"S0022481200053767_ref006","doi-asserted-by":"publisher","DOI":"10.1007\/BF01201343"},{"key":"S0022481200053767_ref009","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(71)90015-5"},{"key":"S0022481200053767_ref003","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(72)90013-7"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200053767","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,29]],"date-time":"2019-05-29T15:40:07Z","timestamp":1559144407000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200053767\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1975,6]]},"references-count":12,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1975,6]]}},"alternative-id":["S0022481200053767"],"URL":"https:\/\/doi.org\/10.2307\/2271901","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1975,6]]}}}