{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T15:00:52Z","timestamp":1777474852534,"version":"3.51.4"},"reference-count":0,"publisher":"Walter de Gruyter GmbH","issue":"4","license":[{"start":{"date-parts":[[2014,12,1]],"date-time":"2014-12-01T00:00:00Z","timestamp":1417392000000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-sa\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2014,12,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>In this article, we formalize a torsion Z-module and a torsionfree Z-module. Especially, we prove formally that finitely generated torsion-free Z-modules are finite rank free. We also formalize properties related to rank of finite rank free Z-modules. The notion of Z-module is necessary for solving lattice problems, LLL (Lenstra, Lenstra, and Lov\u00b4asz) base reduction algorithm [20], cryptographic systems with lattice [21], and coding theory [11].<\/jats:p>","DOI":"10.2478\/forma-2014-0028","type":"journal-article","created":{"date-parts":[[2015,6,16]],"date-time":"2015-06-16T13:39:57Z","timestamp":1434461997000},"page":"277-289","source":"Crossref","is-referenced-by-count":0,"title":["Torsion Z-module and Torsion-free Z-module"],"prefix":"10.2478","volume":"22","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":"Kazuhisa","family":"Nakasho","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":[[2014,12,31]]},"container-title":["Formalized Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/content.sciendo.com\/view\/journals\/forma\/22\/4\/article-p277.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/reference-global.com\/pdf\/10.2478\/forma-2014-0028","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T13:50:28Z","timestamp":1777384228000},"score":1,"resource":{"primary":{"URL":"https:\/\/reference-global.com\/article\/10.2478\/forma-2014-0028"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,12,1]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2014,12,31]]},"published-print":{"date-parts":[[2014,12,1]]}},"alternative-id":["10.2478\/forma-2014-0028"],"URL":"https:\/\/doi.org\/10.2478\/forma-2014-0028","relation":{},"ISSN":["1898-9934"],"issn-type":[{"value":"1898-9934","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,12,1]]}}}