{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,10,31]],"date-time":"2022-10-31T17:31:00Z","timestamp":1667237460532},"reference-count":0,"publisher":"Walter de Gruyter GmbH","issue":"1","license":[{"start":{"date-parts":[[2016,3,1]],"date-time":"2016-03-01T00:00:00Z","timestamp":1456790400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by-sa\/3.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2016,3,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n               <jats:p> In this article, we formalize the definition of divisible \u2124-module and its properties in the Mizar system [3]. We formally prove that any non-trivial divisible \u2124-modules are not finitely-generated.We introduce a divisible \u2124-module, equivalent to a vector space of a torsion-free \u2124-module with a coefficient ring \u211a. \u2124-modules are important for lattice problems, LLL (Lenstra, Lenstra and Lov\u00e1sz) base reduction algorithm [15], cryptographic systems with lattices [16] and coding theory [8].<\/jats:p>","DOI":"10.1515\/forma-2016-0004","type":"journal-article","created":{"date-parts":[[2016,8,22]],"date-time":"2016-08-22T14:36:23Z","timestamp":1471876583000},"page":"37-47","source":"Crossref","is-referenced-by-count":4,"title":["Divisible \u2124-modules"],"prefix":"10.1515","volume":"24","author":[{"given":"Yuichi","family":"Futa","sequence":"first","affiliation":[{"name":"Japan Advanced Institute of Science and Technology Ishikawa, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yasunari","family":"Shidama","sequence":"additional","affiliation":[{"name":"Shinshu University Nagano, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2016,8,31]]},"container-title":["Formalized Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/content.sciendo.com\/view\/journals\/forma\/24\/1\/article-p37.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.sciendo.com\/article\/10.1515\/forma-2016-0004","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,4,7]],"date-time":"2021-04-07T00:13:09Z","timestamp":1617754389000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.sciendo.com\/article\/10.1515\/forma-2016-0004"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,3,1]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2016,8,31]]},"published-print":{"date-parts":[[2016,3,1]]}},"alternative-id":["10.1515\/forma-2016-0004"],"URL":"https:\/\/doi.org\/10.1515\/forma-2016-0004","relation":{},"ISSN":["1898-9934"],"issn-type":[{"value":"1898-9934","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,3,1]]}}}