{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,2]],"date-time":"2022-04-02T15:20:49Z","timestamp":1648912849133},"reference-count":0,"publisher":"Walter de Gruyter GmbH","issue":"4","license":[{"start":{"date-parts":[[2015,12,1]],"date-time":"2015-12-01T00:00:00Z","timestamp":1448928000000},"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":[[2015,12,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n               <jats:p>In this article, we formalize in Mizar [7] the definition of \u201ctorsion part\u201d of \u2124-module and its properties. We show \u2124-module generated by the field of rational numbers as an example of torsion-free non free \u2124-modules. We also formalize the rank-nullity theorem over finite-rank free \u2124-modules (previously formalized in [1]). \u2124-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lov\u00e1sz) base reduction algorithm [23] and cryptographic systems with lattices [24].<\/jats:p>","DOI":"10.1515\/forma-2015-0024","type":"journal-article","created":{"date-parts":[[2016,3,30]],"date-time":"2016-03-30T23:40:48Z","timestamp":1459381248000},"page":"297-307","source":"Crossref","is-referenced-by-count":1,"title":["Torsion Part of \u2124-module"],"prefix":"10.1515","volume":"23","author":[{"given":"Yuichi","family":"Futa","sequence":"first","affiliation":[{"name":"Japan Advanced Institute of Science and Technology, Ishikawa, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hiroyuki","family":"Okazaki","sequence":"additional","affiliation":[{"name":"Shinshu University, Nagano, 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":[[2016,3,25]]},"container-title":["Formalized Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/content.sciendo.com\/view\/journals\/forma\/23\/4\/article-p297.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.sciendo.com\/article\/10.1515\/forma-2015-0024","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,4,7]],"date-time":"2021-04-07T08:17:09Z","timestamp":1617783429000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.sciendo.com\/article\/10.1515\/forma-2015-0024"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,12,1]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2016,3,25]]},"published-print":{"date-parts":[[2015,12,1]]}},"alternative-id":["10.1515\/forma-2015-0024"],"URL":"https:\/\/doi.org\/10.1515\/forma-2015-0024","relation":{},"ISSN":["1898-9934","1426-2630"],"issn-type":[{"value":"1898-9934","type":"electronic"},{"value":"1426-2630","type":"print"}],"subject":[],"published":{"date-parts":[[2015,12,1]]}}}