{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,8]],"date-time":"2026-04-08T10:27:29Z","timestamp":1775644049467,"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":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":"This paper is mainly concerned with describing complete theories of modules by decomposing them (up to elementary equivalence) into direct products of simpler modules. In \u00a71, I give a decomposition theorem which works for arbitrary direct product theories T<\/jats:italic>. Given such a T<\/jats:italic>, I define T<\/jats:italic>-indecomposable structures and show that every model of T<\/jats:italic> is elementarily equivalent to a direct product of T<\/jats:italic>-indecomposable models of T<\/jats:italic>. In \u00a72, I show that if R<\/jats:italic> is a commutative ring then every R<\/jats:italic>-module is elementarily equivalent to \u03a0M<\/jats:italic><\/jats:sub>M<\/jats:italic><\/jats:sub> where M<\/jats:italic> ranges over the maximal ideals of R<\/jats:italic> and M<\/jats:italic><\/jats:sub> is the localization of at M<\/jats:italic>. This is applied to prove that if R<\/jats:italic> is a commutative von Neumann regular ring and TR<\/jats:sub><\/jats:italic> is the theory of R<\/jats:italic>-modules then the TR<\/jats:sub><\/jats:italic>-indecomposables are precisely the cyclic modules of the form R<\/jats:italic>\/M<\/jats:italic> where M<\/jats:italic> is a maximal ideal. In \u00a73, I use the decomposition established in \u00a72 to characterize the \u03c91<\/jats:sub>-categorical and \u03c9-stable modules over a countable commutative von Neumann regular ring and the superstable modules over a commutative von Neumann regular ring of arbitrary cardinality. In the process, I also prove several general characterizations of \u03c9-stable and superstable modules; e.g., if R<\/jats:italic> is any countable ring, then an R<\/jats:italic>-moduIe is \u03c9-stable if and only if every R<\/jats:italic>-module elementarily equivalent to it is equationally compact.<\/jats:p>","DOI":"10.2307\/2273705","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:49:34Z","timestamp":1146937774000},"page":"77-88","source":"Crossref","is-referenced-by-count":16,"title":["Direct product decomposition of theories of modules"],"prefix":"10.1017","volume":"44","author":[{"given":"Steven","family":"Garavaglia","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200048738_ref005","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(71)90016-7"},{"key":"S0022481200048738_ref011","volume-title":"Algebra","author":"Lang","year":"1971"},{"key":"S0022481200048738_ref016","volume-title":"An introduction to homological algebra","author":"Northcott","year":"1966"},{"key":"S0022481200048738_ref013","doi-asserted-by":"publisher","DOI":"10.4064\/fm-70-3-253-270"},{"key":"S0022481200048738_ref006","doi-asserted-by":"publisher","DOI":"10.4064\/fm-47-1-57-103"},{"key":"S0022481200048738_ref012","volume-title":"Finite rings with identity","author":"McDonald","year":"1974"},{"key":"S0022481200048738_ref014","doi-asserted-by":"publisher","DOI":"10.1007\/BF01668811"},{"key":"S0022481200048738_ref009","volume-title":"Multiplicative ideal theory","author":"Gilmer","year":"1972"},{"key":"S0022481200048738_ref003","first-page":"335","volume":"37","author":"Eklof","year":"1972","journal-title":"Some model theory of abelian groups"},{"key":"S0022481200048738_ref007","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(70)90002-1"},{"key":"S0022481200048738_ref004","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(72)90013-7"},{"key":"S0022481200048738_ref008","unstructured":"Garavaglia S. , Elementary equivalence of modules, unpublished."},{"key":"S0022481200048738_ref010","volume-title":"Commutative rings","author":"Kaplansky","year":"1970"},{"key":"S0022481200048738_ref015","unstructured":"Monk L. , Elementary-recursive decision procedures, Ph.D. dissertation, University of California, Berkeley, 1975."},{"key":"S0022481200048738_ref001","unstructured":"Baur W. , Elimination of quantifiers for modules (to appear)."},{"key":"S0022481200048738_ref017","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, Serie A"},{"key":"S0022481200048738_ref002","doi-asserted-by":"publisher","DOI":"10.1007\/BF01179851"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200048738","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,26]],"date-time":"2019-05-26T17:56:42Z","timestamp":1558893402000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200048738\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1979,3]]},"references-count":17,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1979,3]]}},"alternative-id":["S0022481200048738"],"URL":"https:\/\/doi.org\/10.2307\/2273705","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1979,3]]}}}