{"id":"https://openalex.org/W2508409595","doi":"https://doi.org/10.1515/forma-2016-0004","title":"Divisible \u2124-modules","display_name":"Divisible \u2124-modules","publication_year":2016,"publication_date":"2016-03-01","ids":{"openalex":"https://openalex.org/W2508409595","doi":"https://doi.org/10.1515/forma-2016-0004","mag":"2508409595"},"language":"en","primary_location":{"id":"doi:10.1515/forma-2016-0004","is_oa":true,"landing_page_url":"https://doi.org/10.1515/forma-2016-0004","pdf_url":"https://content.sciendo.com/downloadpdf/journals/forma/24/1/article-p37.pdf","source":{"id":"https://openalex.org/S179241345","display_name":"Formalized Mathematics","issn_l":"1426-2630","issn":["1426-2630","1898-9934"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320322","host_organization_name":"De Gruyter Open","host_organization_lineage":["https://openalex.org/P4310320322","https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter Open","De Gruyter"],"type":"journal"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formalized Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://content.sciendo.com/downloadpdf/journals/forma/24/1/article-p37.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5076839736","display_name":"Yuichi Futa","orcid":null},"institutions":[{"id":"https://openalex.org/I177738480","display_name":"Japan Advanced Institute of Science and Technology","ror":"https://ror.org/03frj4r98","country_code":"JP","type":"education","lineage":["https://openalex.org/I177738480"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Yuichi Futa","raw_affiliation_strings":["Japan Advanced Institute of Science and Technology Ishikawa, Japan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Japan Advanced Institute of Science and Technology Ishikawa, Japan","institution_ids":["https://openalex.org/I177738480"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5055549852","display_name":"Yasunari Shidama","orcid":"https://orcid.org/0000-0001-5634-0838"},"institutions":[{"id":"https://openalex.org/I137975476","display_name":"Shinshu University","ror":"https://ror.org/0244rem06","country_code":"JP","type":"education","lineage":["https://openalex.org/I137975476"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Yasunari Shidama","raw_affiliation_strings":["Shinshu University Nagano, Japan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Shinshu University Nagano, Japan","institution_ids":["https://openalex.org/I137975476"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.8833,"has_fulltext":true,"cited_by_count":5,"citation_normalized_percentile":{"value":0.84267805,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":96},"biblio":{"volume":"24","issue":"1","first_page":"37","last_page":"47"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10237","display_name":"Cryptography and Data Security","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10237","display_name":"Cryptography and Data Security","score":0.9998000264167786,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":0.9997000098228455,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11797","display_name":"graph theory and CDMA systems","score":0.9993000030517578,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7961339950561523},{"id":"https://openalex.org/keywords/cryptography","display_name":"Cryptography","score":0.5751278400421143},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4572162330150604},{"id":"https://openalex.org/keywords/lattice","display_name":"Lattice (music)","score":0.43951383233070374},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.3214450478553772},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.19206342101097107}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7961339950561523},{"id":"https://openalex.org/C178489894","wikidata":"https://www.wikidata.org/wiki/Q8789","display_name":"Cryptography","level":2,"score":0.5751278400421143},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4572162330150604},{"id":"https://openalex.org/C2781204021","wikidata":"https://www.wikidata.org/wiki/Q6497091","display_name":"Lattice (music)","level":2,"score":0.43951383233070374},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3214450478553772},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.19206342101097107},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1515/forma-2016-0004","is_oa":true,"landing_page_url":"https://doi.org/10.1515/forma-2016-0004","pdf_url":"https://content.sciendo.com/downloadpdf/journals/forma/24/1/article-p37.pdf","source":{"id":"https://openalex.org/S179241345","display_name":"Formalized Mathematics","issn_l":"1426-2630","issn":["1426-2630","1898-9934"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320322","host_organization_name":"De Gruyter Open","host_organization_lineage":["https://openalex.org/P4310320322","https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter Open","De Gruyter"],"type":"journal"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formalized Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:6f220ff1bbaf48d4bb15f9a17ff5e994","is_oa":false,"landing_page_url":"https://doaj.org/article/6f220ff1bbaf48d4bb15f9a17ff5e994","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Formalized Mathematics, Vol 24, Iss 1, Pp 37-47 (2016)","raw_type":"article"},{"id":"pmh:oai:repozytorium.uwb.edu.pl:11320/5489","is_oa":false,"landing_page_url":"http://hdl.handle.net/11320/5489","pdf_url":null,"source":{"id":"https://openalex.org/S4306400933","display_name":"The Repository of University in Bialystok (University in Bialystok)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210094198","host_organization_name":"University Clinical Hospital In Bialystok","host_organization_lineage":["https://openalex.org/I4210094198"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Article"}],"best_oa_location":{"id":"doi:10.1515/forma-2016-0004","is_oa":true,"landing_page_url":"https://doi.org/10.1515/forma-2016-0004","pdf_url":"https://content.sciendo.com/downloadpdf/journals/forma/24/1/article-p37.pdf","source":{"id":"https://openalex.org/S179241345","display_name":"Formalized Mathematics","issn_l":"1426-2630","issn":["1426-2630","1898-9934"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320322","host_organization_name":"De Gruyter Open","host_organization_lineage":["https://openalex.org/P4310320322","https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter Open","De Gruyter"],"type":"journal"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formalized Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2508409595.pdf","grobid_xml":"https://content.openalex.org/works/W2508409595.grobid-xml"},"referenced_works_count":17,"referenced_works":["https://openalex.org/W180239714","https://openalex.org/W192809544","https://openalex.org/W260340778","https://openalex.org/W1259055132","https://openalex.org/W1485280854","https://openalex.org/W2026904929","https://openalex.org/W2075354892","https://openalex.org/W2105559527","https://openalex.org/W2138253682","https://openalex.org/W2139831914","https://openalex.org/W2141040012","https://openalex.org/W2159772409","https://openalex.org/W2295728669","https://openalex.org/W2326370346","https://openalex.org/W2777805251","https://openalex.org/W4253086862","https://openalex.org/W4285719527"],"related_works":["https://openalex.org/W1979597421","https://openalex.org/W2007980826","https://openalex.org/W4245490552","https://openalex.org/W2061531152","https://openalex.org/W3002753104","https://openalex.org/W2077600819","https://openalex.org/W1587224694","https://openalex.org/W2042127053","https://openalex.org/W2142036596","https://openalex.org/W4238204885"],"abstract_inverted_index":{"Summary":[0],"In":[1],"this":[2],"article,":[3],"we":[4],"formalize":[5],"the":[6,15],"definition":[7],"of":[8,39],"divisible":[9,25,32],"\u2124-module":[10,42],"and":[11,57,68],"its":[12],"properties":[13],"in":[14],"Mizar":[16],"system":[17],"[3].":[18],"We":[19],"formally":[20],"prove":[21],"that":[22],"any":[23],"non-trivial":[24],"\u2124-modules":[26,48],"are":[27,49],"not":[28],"finitely-generated.We":[29],"introduce":[30],"a":[31,36,40,44],"\u2124-module,":[33],"equivalent":[34],"to":[35],"vector":[37],"space":[38],"torsion-free":[41],"with":[43,65],"coefficient":[45],"ring":[46],"\u211a.":[47],"important":[50],"for":[51],"lattice":[52],"problems,":[53],"LLL":[54],"(Lenstra,":[55],"Lenstra":[56],"Lov\u00e1sz)":[58],"base":[59],"reduction":[60],"algorithm":[61],"[15],":[62],"cryptographic":[63],"systems":[64],"lattices":[66],"[16]":[67],"coding":[69],"theory":[70],"[8].":[71]},"counts_by_year":[{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":2},{"year":2017,"cited_by_count":2}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}
