{"id":"https://openalex.org/W2084432067","doi":"https://doi.org/10.2307/2273039","title":"The word problem for division rings","display_name":"The word problem for division rings","publication_year":1973,"publication_date":"1973-09-01","ids":{"openalex":"https://openalex.org/W2084432067","doi":"https://doi.org/10.2307/2273039","mag":"2084432067"},"language":"en","primary_location":{"id":"doi:10.2307/2273039","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2273039","pdf_url":null,"source":{"id":"https://openalex.org/S177675236","display_name":"Journal of Symbolic Logic","issn_l":"0022-4812","issn":["0022-4812","1943-5886"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Symbolic Logic","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5046795804","display_name":"Angus Macintyre","orcid":"https://orcid.org/0000-0003-4740-5710"},"institutions":[{"id":"https://openalex.org/I144852770","display_name":"The King's College","ror":"https://ror.org/051j50p12","country_code":"US","type":"education","lineage":["https://openalex.org/I144852770"]},{"id":"https://openalex.org/I4210153400","display_name":"King's College Hospital","ror":"https://ror.org/044nptt90","country_code":"GB","type":"healthcare","lineage":["https://openalex.org/I4210111135","https://openalex.org/I4210153400"]}],"countries":["GB","US"],"is_corresponding":true,"raw_author_name":"Angus Macintyre","raw_affiliation_strings":["King's College, Old Aberdeen, Scotland"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"King's College, Old Aberdeen, Scotland","institution_ids":["https://openalex.org/I144852770","https://openalex.org/I4210153400"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5046795804"],"corresponding_institution_ids":["https://openalex.org/I144852770","https://openalex.org/I4210153400"],"apc_list":null,"apc_paid":null,"fwci":0.9272,"has_fulltext":false,"cited_by_count":20,"citation_normalized_percentile":{"value":0.76853253,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":96},"biblio":{"volume":"38","issue":"3","first_page":"428","last_page":"436"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10849","display_name":"Finite Group Theory Research","score":0.9894999861717224,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10849","display_name":"Finite Group Theory Research","score":0.9894999861717224,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10304","display_name":"Geometric and Algebraic Topology","score":0.9879999756813049,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10287","display_name":"Algebraic structures and combinatorial models","score":0.987500011920929,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/division","display_name":"Division (mathematics)","score":0.8405591249465942},{"id":"https://openalex.org/keywords/word-problem","display_name":"Word problem (mathematics education)","score":0.6723869442939758},{"id":"https://openalex.org/keywords/word","display_name":"Word (group theory)","score":0.6567384600639343},{"id":"https://openalex.org/keywords/group","display_name":"Group (periodic table)","score":0.635193943977356},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6048475503921509},{"id":"https://openalex.org/keywords/division-ring","display_name":"Division ring","score":0.5801162719726562},{"id":"https://openalex.org/keywords/degree","display_name":"Degree (music)","score":0.4492158889770508},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.44205495715141296},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.38235267996788025},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.3710382580757141},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3534785509109497},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.35124197602272034},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.060314446687698364}],"concepts":[{"id":"https://openalex.org/C60798267","wikidata":"https://www.wikidata.org/wiki/Q1226939","display_name":"Division (mathematics)","level":2,"score":0.8405591249465942},{"id":"https://openalex.org/C90988201","wikidata":"https://www.wikidata.org/wiki/Q255912","display_name":"Word problem (mathematics education)","level":2,"score":0.6723869442939758},{"id":"https://openalex.org/C90805587","wikidata":"https://www.wikidata.org/wiki/Q10944557","display_name":"Word (group theory)","level":2,"score":0.6567384600639343},{"id":"https://openalex.org/C2781311116","wikidata":"https://www.wikidata.org/wiki/Q83306","display_name":"Group (periodic table)","level":2,"score":0.635193943977356},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6048475503921509},{"id":"https://openalex.org/C31671303","wikidata":"https://www.wikidata.org/wiki/Q650741","display_name":"Division ring","level":3,"score":0.5801162719726562},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.4492158889770508},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.44205495715141296},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.38235267996788025},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.3710382580757141},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3534785509109497},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.35124197602272034},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.060314446687698364},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C178790620","wikidata":"https://www.wikidata.org/wiki/Q11351","display_name":"Organic chemistry","level":1,"score":0.0},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","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":2,"locations":[{"id":"doi:10.2307/2273039","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2273039","pdf_url":null,"source":{"id":"https://openalex.org/S177675236","display_name":"Journal of Symbolic Logic","issn_l":"0022-4812","issn":["0022-4812","1943-5886"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Symbolic Logic","raw_type":"journal-article"},{"id":"pmh:oai:CULeuclid:euclid.jsl/1183738755","is_oa":false,"landing_page_url":"http://projecteuclid.org/euclid.jsl/1183738755","pdf_url":null,"source":{"id":"https://openalex.org/S4306400787","display_name":"Project Euclid (Cornell University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Text"}],"best_oa_location":null,"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/16","display_name":"Peace, Justice and strong institutions","score":0.4699999988079071}],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":24,"referenced_works":["https://openalex.org/W1558352842","https://openalex.org/W1573036064","https://openalex.org/W1578424393","https://openalex.org/W1967684312","https://openalex.org/W2000141173","https://openalex.org/W2000947982","https://openalex.org/W2001162243","https://openalex.org/W2008907564","https://openalex.org/W2034460536","https://openalex.org/W2036033272","https://openalex.org/W2069128211","https://openalex.org/W2074392054","https://openalex.org/W2136111664","https://openalex.org/W2316365877","https://openalex.org/W2328824725","https://openalex.org/W2333555861","https://openalex.org/W2529932703","https://openalex.org/W2530520115","https://openalex.org/W2554842236","https://openalex.org/W2561618607","https://openalex.org/W2594961821","https://openalex.org/W2755351717","https://openalex.org/W4238491266","https://openalex.org/W4246894044"],"related_works":["https://openalex.org/W2022472189","https://openalex.org/W2070939056","https://openalex.org/W4294168466","https://openalex.org/W2379498264","https://openalex.org/W1565833102","https://openalex.org/W2264352149","https://openalex.org/W4223469647","https://openalex.org/W4229596634","https://openalex.org/W4300430055","https://openalex.org/W365049119"],"abstract_inverted_index":{"In":[0],"this":[1,97],"paper":[2],"we":[3,100],"prove":[4],"that":[5,99,102],"the":[6,19,72,103],"word":[7,44,104],"problem":[8,45,105],"for":[9,22,46,87,106],"division":[10,41,107],"rings":[11,108],"is":[12,48,79,93],"recursively":[13],"unsolvable.":[14],"Our":[15],"proof":[16],"relies":[17],"on":[18,40],"corresponding":[20],"result":[21],"groups":[23,47],"[7],":[24,62],"[28],":[25,64],"and":[26,91],"makes":[27,85],"essential":[28],"use":[29],"of":[30,53,71,76],"P.":[31],"M.":[32],"Cohn's":[33],"recent":[34],"work":[35],"[11],":[36],"[13],":[37],"[15],":[38],"[16]":[39],"rings.":[42],"The":[43],"usually":[49],"formulated":[50],"in":[51,61,69,81],"terms":[52,70],"group":[54,77],"presentations":[55],"or":[56],"finitely":[57],"presented":[58],"groups,":[59],"as":[60],"[24],":[63],"[30].":[65],"An":[66],"equivalent":[67],"formulation,":[68],"universal":[73],"Horn":[74],"sentences":[75],"theory,":[78],"mentioned":[80],"[32].":[82],"This":[83],"formulation":[84,98],"sense":[86],"arbitrary":[88],"first-order":[89],"theories,":[90],"it":[92],"with":[94],"respect":[95],"to":[96],"show":[101],"has":[109],"degree":[110],"0\u2032.":[111]},"counts_by_year":[{"year":2023,"cited_by_count":1},{"year":2017,"cited_by_count":1},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}