{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,18]],"date-time":"2025-10-18T00:27:26Z","timestamp":1760747246932,"version":"build-2065373602"},"reference-count":13,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2025,5,21]],"date-time":"2025-05-21T00:00:00Z","timestamp":1747785600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Proceedings of the Edinburgh Mathematical Society"],"published-print":{"date-parts":[[2025,11]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>This paper is a continuation of a project to determine which skew polynomial algebras <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0013091525000161_inline3.png\"\/><jats:tex-math>$S = R[\\theta; \\alpha]$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> satisfy property <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0013091525000161_inline4.png\"\/><jats:tex-math>$(\\diamond)$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, namely that the injective hull of every simple <jats:italic>S<\/jats:italic>-module is locally Artinian, where <jats:italic>k<\/jats:italic> is a field, <jats:italic>R<\/jats:italic> is a commutative Noetherian <jats:italic>k<\/jats:italic>-algebra and <jats:italic>\u03b1<\/jats:italic> is a <jats:italic>k<\/jats:italic>-algebra automorphism of <jats:italic>R<\/jats:italic>. Earlier work (which we review) and further analysis done here lead us to focus on the case where <jats:italic>S<\/jats:italic> is a primitive domain and <jats:italic>R<\/jats:italic> has Krull dimension 1 and contains an uncountable field. Then we show first that if <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0013091525000161_inline5.png\"\/><jats:tex-math>$|\\mathrm{Spec}(R)|$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is infinite then <jats:italic>S<\/jats:italic> does not satisfy <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0013091525000161_inline6.png\"\/><jats:tex-math>$(\\diamond)$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. Secondly, we show that when <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0013091525000161_inline7.png\"\/><jats:tex-math>$R = k[X]_{ \\lt X \\gt }$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0013091525000161_inline8.png\"\/><jats:tex-math>$\\alpha (X) = qX$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> where <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0013091525000161_inline9.png\"\/><jats:tex-math>$q \\in k \\setminus \\{0\\}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is not a root of unity then <jats:italic>S<\/jats:italic> does not satisfy <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0013091525000161_inline10.png\"\/><jats:tex-math>$(\\diamond)$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. This is in complete contrast to our earlier result that, when <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0013091525000161_inline11.png\"\/><jats:tex-math>$R = k[[X]]$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> and <jats:italic>\u03b1<\/jats:italic> is an arbitrary <jats:italic>k<\/jats:italic>-algebra automorphism of infinite order, <jats:italic>S<\/jats:italic> satisfies <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0013091525000161_inline12.png\"\/><jats:tex-math>$(\\diamond)$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. A number of open questions are stated.<\/jats:p>","DOI":"10.1017\/s0013091525000161","type":"journal-article","created":{"date-parts":[[2025,5,21]],"date-time":"2025-05-21T04:52:38Z","timestamp":1747803158000},"page":"1169-1187","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["Property (\u25c7) for Ore extensions of small Krull dimension"],"prefix":"10.1017","volume":"68","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3497-3372","authenticated-orcid":false,"given":"Ken","family":"Brown","sequence":"first","affiliation":[{"name":"University of Glasgow"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8487-7541","authenticated-orcid":false,"given":"Paula A. A. B.","family":"Carvalho","sequence":"additional","affiliation":[{"name":"Universidade do Porto"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8749-6229","authenticated-orcid":false,"given":"Jerzy","family":"Matczuk","sequence":"additional","affiliation":[{"name":"Institute of Mathematics, University of Warsaw"}]}],"member":"56","published-online":{"date-parts":[[2025,5,21]]},"reference":[{"key":"S0013091525000161_ref11","doi-asserted-by":"publisher","DOI":"10.1090\/conm\/562\/11138"},{"key":"S0013091525000161_ref3","doi-asserted-by":"publisher","DOI":"10.1007\/s00209-018-2128-8"},{"key":"S0013091525000161_ref1","doi-asserted-by":"publisher","DOI":"10.1006\/jabr.1997.7038"},{"key":"S0013091525000161_ref6","doi-asserted-by":"publisher","DOI":"10.1016\/0021-8693(85)90148-6"},{"key":"S0013091525000161_ref7","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1986-0835871-0"},{"key":"S0013091525000161_ref5","doi-asserted-by":"publisher","DOI":"10.1017\/S0017089510000261"},{"key":"S0013091525000161_ref8","doi-asserted-by":"publisher","DOI":"10.1016\/0021-8693(79)90341-7"},{"key":"S0013091525000161_ref9","doi-asserted-by":"publisher","DOI":"10.1080\/00927879608825699"},{"key":"S0013091525000161_ref4","doi-asserted-by":"publisher","DOI":"10.1016\/j.jpaa.2023.107441"},{"key":"S0013091525000161_ref12","doi-asserted-by":"publisher","DOI":"10.1007\/BF01558598"},{"key":"S0013091525000161_ref10","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1958.8.511"},{"key":"S0013091525000161_ref13","doi-asserted-by":"publisher","DOI":"10.1007\/BF01237858"},{"volume-title":"Volume CRM Barcelona of Advanced Courses in Mathematics","year":"2002","author":"Brown","key":"S0013091525000161_ref2"}],"container-title":["Proceedings of the Edinburgh Mathematical Society"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0013091525000161","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,17]],"date-time":"2025-10-17T15:33:04Z","timestamp":1760715184000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0013091525000161\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,5,21]]},"references-count":13,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2025,11]]}},"alternative-id":["S0013091525000161"],"URL":"https:\/\/doi.org\/10.1017\/s0013091525000161","relation":{},"ISSN":["0013-0915","1464-3839"],"issn-type":[{"type":"print","value":"0013-0915"},{"type":"electronic","value":"1464-3839"}],"subject":[],"published":{"date-parts":[[2025,5,21]]},"assertion":[{"value":"\u00a9 The Author(s), 2025. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}}]}}