{"id":"https://openalex.org/W1991984812","doi":"https://doi.org/10.1002/malq.200410018","title":"If vector spaces are projective modules then multiple choice holds","display_name":"If vector spaces are projective modules then multiple choice holds","publication_year":2005,"publication_date":"2005-01-26","ids":{"openalex":"https://openalex.org/W1991984812","doi":"https://doi.org/10.1002/malq.200410018","mag":"1991984812"},"language":"en","primary_location":{"id":"doi:10.1002/malq.200410018","is_oa":false,"landing_page_url":"https://doi.org/10.1002/malq.200410018","pdf_url":null,"source":{"id":"https://openalex.org/S172351290","display_name":"Mathematical logic quarterly","issn_l":"0942-5616","issn":["0942-5616","1521-3870"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"MLQ","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/A5108590548","display_name":"Paul C. Howard","orcid":null},"institutions":[{"id":"https://openalex.org/I138216421","display_name":"Eastern Michigan University","ror":"https://ror.org/02ehshm78","country_code":"US","type":"education","lineage":["https://openalex.org/I138216421"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Paul Howard","raw_affiliation_strings":["Department of Mathematics, Eastern Michigan University, Ypsilanti, MI 48197, U. S. A","Department of Mathematics, Eastern Michigan University, Ypsilanti, MI 48197, U, S.A"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Eastern Michigan University, Ypsilanti, MI 48197, U. S. A","institution_ids":["https://openalex.org/I138216421"]},{"raw_affiliation_string":"Department of Mathematics, Eastern Michigan University, Ypsilanti, MI 48197, U, S.A","institution_ids":["https://openalex.org/I138216421"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5108590548"],"corresponding_institution_ids":["https://openalex.org/I138216421"],"apc_list":{"value":2630,"currency":"USD","value_usd":2630},"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.10803894,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"51","issue":"2","first_page":"187","last_page":"190"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11727","display_name":"Advanced Algebra and Logic","score":0.9703999757766724,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"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/T11727","display_name":"Advanced Algebra and Logic","score":0.9703999757766724,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"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/T11703","display_name":"Rings, Modules, and Algebras","score":0.9700999855995178,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"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/mathematics","display_name":"Mathematics","score":0.7580322027206421},{"id":"https://openalex.org/keywords/projective-test","display_name":"Projective test","score":0.7234888076782227},{"id":"https://openalex.org/keywords/assertion","display_name":"Assertion","score":0.693575918674469},{"id":"https://openalex.org/keywords/vector-space","display_name":"Vector space","score":0.615208625793457},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.5450960397720337},{"id":"https://openalex.org/keywords/projective-space","display_name":"Projective space","score":0.5409747958183289},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.5327135920524597},{"id":"https://openalex.org/keywords/ordered-vector-space","display_name":"Ordered vector space","score":0.5048059225082397},{"id":"https://openalex.org/keywords/space","display_name":"Space (punctuation)","score":0.5010092258453369},{"id":"https://openalex.org/keywords/axiom","display_name":"Axiom","score":0.46917277574539185},{"id":"https://openalex.org/keywords/axiom-of-choice","display_name":"Axiom of choice","score":0.42944595217704773},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.36173880100250244},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.32585376501083374},{"id":"https://openalex.org/keywords/set-theory","display_name":"Set theory","score":0.28393474221229553},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.1891472339630127},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.10941770672798157},{"id":"https://openalex.org/keywords/interpolation-space","display_name":"Interpolation space","score":0.05971038341522217},{"id":"https://openalex.org/keywords/programming-language","display_name":"Programming language","score":0.05719617009162903}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7580322027206421},{"id":"https://openalex.org/C177846678","wikidata":"https://www.wikidata.org/wiki/Q1501864","display_name":"Projective test","level":2,"score":0.7234888076782227},{"id":"https://openalex.org/C40422974","wikidata":"https://www.wikidata.org/wiki/Q741248","display_name":"Assertion","level":2,"score":0.693575918674469},{"id":"https://openalex.org/C13336665","wikidata":"https://www.wikidata.org/wiki/Q125977","display_name":"Vector space","level":2,"score":0.615208625793457},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.5450960397720337},{"id":"https://openalex.org/C75280867","wikidata":"https://www.wikidata.org/wiki/Q877775","display_name":"Projective space","level":3,"score":0.5409747958183289},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.5327135920524597},{"id":"https://openalex.org/C113837420","wikidata":"https://www.wikidata.org/wiki/Q1503340","display_name":"Ordered vector space","level":5,"score":0.5048059225082397},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.5010092258453369},{"id":"https://openalex.org/C167729594","wikidata":"https://www.wikidata.org/wiki/Q17736","display_name":"Axiom","level":2,"score":0.46917277574539185},{"id":"https://openalex.org/C151797676","wikidata":"https://www.wikidata.org/wiki/Q179692","display_name":"Axiom of choice","level":4,"score":0.42944595217704773},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.36173880100250244},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.32585376501083374},{"id":"https://openalex.org/C153046414","wikidata":"https://www.wikidata.org/wiki/Q12482","display_name":"Set theory","level":3,"score":0.28393474221229553},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.1891472339630127},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.10941770672798157},{"id":"https://openalex.org/C6183050","wikidata":"https://www.wikidata.org/wiki/Q3058180","display_name":"Interpolation space","level":4,"score":0.05971038341522217},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.05719617009162903},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0},{"id":"https://openalex.org/C55112680","wikidata":"https://www.wikidata.org/wiki/Q190549","display_name":"Functional analysis","level":3,"score":0.0},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","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}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1002/malq.200410018","is_oa":false,"landing_page_url":"https://doi.org/10.1002/malq.200410018","pdf_url":null,"source":{"id":"https://openalex.org/S172351290","display_name":"Mathematical logic quarterly","issn_l":"0942-5616","issn":["0942-5616","1521-3870"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"MLQ","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":5,"referenced_works":["https://openalex.org/W328650061","https://openalex.org/W423103167","https://openalex.org/W1608084331","https://openalex.org/W1983910091","https://openalex.org/W4235713663"],"related_works":["https://openalex.org/W2204201083","https://openalex.org/W2327962405","https://openalex.org/W3167136439","https://openalex.org/W2017525113","https://openalex.org/W2079030659","https://openalex.org/W2108615711","https://openalex.org/W3160949262","https://openalex.org/W2042334973","https://openalex.org/W2810928779","https://openalex.org/W1996837194"],"abstract_inverted_index":{"We":[0],"show":[1],"that":[2,5,20],"the":[3,14,21,33],"assertion":[4],"every":[6],"vector":[7],"space":[8],"is":[9],"a":[10],"projective":[11],"module":[12],"implies":[13],"axiom":[15],"of":[16,35],"multiple":[17],"choice":[18],"and":[19],"reverse":[22],"implication":[23],"does":[24],"not":[25],"hold":[26],"in":[27],"set":[28],"theory":[29],"weakened":[30],"to":[31],"permit":[32],"existence":[34],"atoms.":[36],"(\u00a9":[37],"2005":[38],"WILEY-VCH":[39],"Verlag":[40],"GmbH":[41],"&":[42],"Co.":[43],"KGaA,":[44],"Weinheim)":[45]},"counts_by_year":[],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}