{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,19]],"date-time":"2026-08-19T19:51:16Z","timestamp":1787169076094,"version":"build-2736575974"},"reference-count":12,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":4210,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2002,9]]},"abstract":"<jats:p>\n                    For every ring\n                    <jats:italic>S<\/jats:italic>\n                    with identity, the (right) Ziegler spectrum of\n                    <jats:italic>S, Zg<\/jats:italic>\n                    <jats:sub>\n                      <jats:italic>s<\/jats:italic>\n                    <\/jats:sub>\n                    , is the set of (isomorphism classes of) indecomposable pure injective (right) S-modules. The Ziegler topology equips\n                    <jats:italic>Zg<\/jats:italic>\n                    <jats:sub>\n                      <jats:italic>s<\/jats:italic>\n                    <\/jats:sub>\n                    with the structure of a topological space. A typical basic open set in this topology is of the form\n                  <\/jats:p>\n                  <jats:p>\n                    <jats:disp-formula>\n                      <jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200009464_eqnU1\"\/>\n                    <\/jats:disp-formula>\n                  <\/jats:p>\n                  <jats:p>\n                    where\n                    <jats:italic>\u03c6<\/jats:italic>\n                    and\n                    <jats:italic>\u03c8<\/jats:italic>\n                    are\n                    <jats:italic>pp<\/jats:italic>\n                    -formulas (with at most one free variable) in the first order language\n                    <jats:italic>\n                      L\n                      <jats:sub>s<\/jats:sub>\n                    <\/jats:italic>\n                    for\n                    <jats:italic>S<\/jats:italic>\n                    -modules; let [\n                    <jats:italic>\u03c6<\/jats:italic>\n                    \/\n                    <jats:italic>\u03c8<\/jats:italic>\n                    ] denote the closed set\n                    <jats:italic>Zg<\/jats:italic>\n                    <jats:sub>\n                      <jats:italic>s<\/jats:italic>\n                    <\/jats:sub>\n                    - (\n                    <jats:italic>\u03c6<\/jats:italic>\n                    \/\n                    <jats:italic>\u03c8<\/jats:italic>\n                    ). There is an alternative way to introduce the Ziegler topology on\n                    <jats:italic>Zg<\/jats:italic>\n                    <jats:sub>\n                      <jats:italic>s<\/jats:italic>\n                    <\/jats:sub>\n                    . For every choice of two f.p. (finitely presented) S-modules\n                    <jats:italic>A, B<\/jats:italic>\n                    and an\n                    <jats:italic>S<\/jats:italic>\n                    -module homomorphism\n                    <jats:italic>f<\/jats:italic>\n                    :\n                    <jats:italic>A<\/jats:italic>\n                    \u2192\n                    <jats:italic>B<\/jats:italic>\n                    , consider the set (\n                    <jats:italic>f<\/jats:italic>\n                    ) of the points\n                    <jats:italic>N<\/jats:italic>\n                    in\n                    <jats:italic>Zg<\/jats:italic>\n                    <jats:sub>\n                      <jats:italic>s<\/jats:italic>\n                    <\/jats:sub>\n                    such that some\n                    <jats:italic>S<\/jats:italic>\n                    -homomorphism\n                    <jats:italic>h<\/jats:italic>\n                    :\n                    <jats:italic>A<\/jats:italic>\n                    \u2192\n                    <jats:italic>N<\/jats:italic>\n                    does not factor through\n                    <jats:italic>f<\/jats:italic>\n                    . Take (\n                    <jats:italic>f<\/jats:italic>\n                    ) as a basic open set. The resulting topology on\n                    <jats:italic>Zg<\/jats:italic>\n                    <jats:sub>\n                      <jats:italic>s<\/jats:italic>\n                    <\/jats:sub>\n                    is, again, the Ziegler topology.\n                  <\/jats:p>\n                  <jats:p>\n                    The algebraic and model-theoretic relevance of the Ziegler topology is discussed in [Z], [P] and in many subsequent papers, including [P1], [P2] and [P3], for instance. Here we are interested in the Ziegler spectrum\n                    <jats:italic>\n                      Zg\n                      <jats:sub>RG<\/jats:sub>\n                    <\/jats:italic>\n                    of a group ring\n                    <jats:italic>RG<\/jats:italic>\n                    , where\n                    <jats:italic>R<\/jats:italic>\n                    is a Dedekind domain of characteristic 0 (for example\n                    <jats:italic>R<\/jats:italic>\n                    could be the ring\n                    <jats:italic>Z<\/jats:italic>\n                    of integers) and\n                    <jats:italic>G<\/jats:italic>\n                    is a finite group. In particular we deal with the\n                    <jats:italic>R<\/jats:italic>\n                    -torsionfree points of\n                    <jats:italic>\n                      Zg\n                      <jats:sub>RG<\/jats:sub>\n                    <\/jats:italic>\n                    .\n                  <\/jats:p>\n                  <jats:p>\n                    The main motivation for this is the study of\n                    <jats:italic>RG<\/jats:italic>\n                    -lattices (i.e., finitely generated\n                    <jats:italic>R<\/jats:italic>\n                    -torsionfree\n                    <jats:italic>RG<\/jats:italic>\n                    -modules).\n                  <\/jats:p>","DOI":"10.2178\/jsl\/1190150153","type":"journal-article","created":{"date-parts":[[2007,12,13]],"date-time":"2007-12-13T14:14:26Z","timestamp":1197555266000},"page":"1126-1140","source":"Crossref","is-referenced-by-count":6,"title":["The torsionfree part of the Ziegler spectrum of\n                    <i>RG<\/i>\n                    when\n                    <i>R<\/i>\n                    is a Dedekind domain and\n                    <i>G<\/i>\n                    is a finite group"],"prefix":"10.1017","volume":"67","author":[{"given":"A.","family":"Marcja","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"M.","family":"Prest","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"C.","family":"Toffalori","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200009464_ref011","doi-asserted-by":"publisher","DOI":"10.1007\/s001530050087"},{"key":"S0022481200009464_ref009","first-page":"445","volume-title":"Advances in algebra and model theory","volume":"9","author":"Rothmaler","year":"1997"},{"key":"S0022481200009464_ref008","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-0348-8426-6_19"},{"key":"S0022481200009464_ref007","doi-asserted-by":"publisher","DOI":"10.1006\/jabr.1998.7472"},{"key":"S0022481200009464_ref005","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511600562"},{"key":"S0022481200009464_ref004","doi-asserted-by":"publisher","DOI":"10.1016\/0022-4049(95)00097-G"},{"key":"S0022481200009464_ref010","first-page":"1429","volume":"63","author":"Toffalori","year":"1997","journal-title":"Wildness implies undecidability for lattices over group rings"},{"key":"S0022481200009464_ref003","volume-title":"Basic algebra II","author":"Jacobson","year":"1980"},{"key":"S0022481200009464_ref002","volume-title":"Methods of representation theory with applications to finite groups and orders I","author":"Curtis","year":"1981"},{"key":"S0022481200009464_ref006","unstructured":"Prest M. , The Zariski spectrum of the category of finitely presented modules, preprint, 1998."},{"key":"S0022481200009464_ref012","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(84)90014-9"},{"key":"S0022481200009464_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/s001530050085"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200009464","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,6]],"date-time":"2019-05-06T20:45:57Z","timestamp":1557175557000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200009464\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,9]]},"references-count":12,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2002,9]]}},"alternative-id":["S0022481200009464"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1190150153","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2002,9]]}}}