{"id":"https://openalex.org/W4395676973","doi":"https://doi.org/10.3233/com-230454","title":"Reifying dynamical algebra: Maximal ideals in countable rings, constructively","display_name":"Reifying dynamical algebra: Maximal ideals in countable rings, constructively","publication_year":2024,"publication_date":"2024-04-26","ids":{"openalex":"https://openalex.org/W4395676973","doi":"https://doi.org/10.3233/com-230454"},"language":"en","primary_location":{"id":"doi:10.3233/com-230454","is_oa":false,"landing_page_url":"https://doi.org/10.3233/com-230454","pdf_url":null,"source":{"id":"https://openalex.org/S4210231662","display_name":"Computability","issn_l":"2211-3568","issn":["2211-3568","2211-3576"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310318577","host_organization_name":"IOS Press","host_organization_lineage":["https://openalex.org/P4310318577"],"host_organization_lineage_names":["IOS Press"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Computability","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/A5025704459","display_name":"Ingo Blechschmidt","orcid":null},"institutions":[{"id":"https://openalex.org/I179225836","display_name":"University of Augsburg","ror":"https://ror.org/03p14d497","country_code":"DE","type":"education","lineage":["https://openalex.org/I179225836"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"Ingo Blechschmidt","raw_affiliation_strings":["Institut f\u00fcr Mathematik, Universit\u00e4t Augsburg, Germany"],"affiliations":[{"raw_affiliation_string":"Institut f\u00fcr Mathematik, Universit\u00e4t Augsburg, Germany","institution_ids":["https://openalex.org/I179225836"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5035502484","display_name":"Peter Schuster","orcid":"https://orcid.org/0000-0002-6831-2057"},"institutions":[{"id":"https://openalex.org/I119439378","display_name":"University of Verona","ror":"https://ror.org/039bp8j42","country_code":"IT","type":"education","lineage":["https://openalex.org/I119439378"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Peter Schuster","raw_affiliation_strings":["Dipartimento di Informatico, Universit\u00e0 di Verona, Italy"],"affiliations":[{"raw_affiliation_string":"Dipartimento di Informatico, Universit\u00e0 di Verona, Italy","institution_ids":["https://openalex.org/I119439378"]}]},{"author_position":"last","author":{"id":null,"display_name":"Peter Schuster","orcid":null},"institutions":[{"id":"https://openalex.org/I119439378","display_name":"University of Verona","ror":"https://ror.org/039bp8j42","country_code":"IT","type":"education","lineage":["https://openalex.org/I119439378"]}],"countries":["IT"],"is_corresponding":true,"raw_author_name":"Peter Schuster","raw_affiliation_strings":["Dipartimento di Informatico, Universit\u00e0 di Verona, Italy"],"affiliations":[{"raw_affiliation_string":"Dipartimento di Informatico, Universit\u00e0 di Verona, Italy","institution_ids":["https://openalex.org/I119439378"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5025704459"],"corresponding_institution_ids":["https://openalex.org/I119439378","https://openalex.org/I179225836"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.07003262,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":"1","last_page":"35"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12002","display_name":"Computability, Logic, AI Algorithms","score":0.9990000128746033,"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/T12002","display_name":"Computability, Logic, AI Algorithms","score":0.9990000128746033,"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":0.9958999752998352,"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"}},{"id":"https://openalex.org/T11151","display_name":"Advanced Topology and Set Theory","score":0.9868999719619751,"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/countable-set","display_name":"Countable set","score":0.7410906553268433},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.567561686038971},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5573601722717285},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.488681823015213}],"concepts":[{"id":"https://openalex.org/C110729354","wikidata":"https://www.wikidata.org/wiki/Q185478","display_name":"Countable set","level":2,"score":0.7410906553268433},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.567561686038971},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5573601722717285},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.488681823015213}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.3233/com-230454","is_oa":false,"landing_page_url":"https://doi.org/10.3233/com-230454","pdf_url":null,"source":{"id":"https://openalex.org/S4210231662","display_name":"Computability","issn_l":"2211-3568","issn":["2211-3568","2211-3576"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310318577","host_organization_name":"IOS Press","host_organization_lineage":["https://openalex.org/P4310318577"],"host_organization_lineage_names":["IOS Press"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Computability","raw_type":"journal-article"},{"id":"pmh:oai:uni-augsburg.opus-bayern.de:118080","is_oa":false,"landing_page_url":"https://opus.bibliothek.uni-augsburg.de/opus4/frontdoor/index/index/docId/118080","pdf_url":null,"source":{"id":"https://openalex.org/S4306400930","display_name":"OPUS (Augsburg University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I119916105","host_organization_name":"Augsburg University","host_organization_lineage":["https://openalex.org/I119916105"],"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":"doc-type:article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":126,"referenced_works":["https://openalex.org/W86610579","https://openalex.org/W162931075","https://openalex.org/W174479171","https://openalex.org/W204380459","https://openalex.org/W580391459","https://openalex.org/W1488848753","https://openalex.org/W1501429467","https://openalex.org/W1505957833","https://openalex.org/W1517302412","https://openalex.org/W1530111769","https://openalex.org/W1556453281","https://openalex.org/W1587781051","https://openalex.org/W1594900487","https://openalex.org/W1608084331","https://openalex.org/W1757945753","https://openalex.org/W1843798910","https://openalex.org/W1959175637","https://openalex.org/W1966278402","https://openalex.org/W1968504752","https://openalex.org/W1969682671","https://openalex.org/W1971479478","https://openalex.org/W1972672801","https://openalex.org/W1972937103","https://openalex.org/W1973390504","https://openalex.org/W1977659025","https://openalex.org/W1979476005","https://openalex.org/W1982516483","https://openalex.org/W1986260727","https://openalex.org/W1988824860","https://openalex.org/W1992572940","https://openalex.org/W1993324392","https://openalex.org/W1994808441","https://openalex.org/W1995877143","https://openalex.org/W2005378310","https://openalex.org/W2007469040","https://openalex.org/W2010337608","https://openalex.org/W2010832370","https://openalex.org/W2011659388","https://openalex.org/W2013311002","https://openalex.org/W2016291981","https://openalex.org/W2022040382","https://openalex.org/W2028119090","https://openalex.org/W2031104901","https://openalex.org/W2031254186","https://openalex.org/W2032868887","https://openalex.org/W2033693919","https://openalex.org/W2041549181","https://openalex.org/W2042984937","https://openalex.org/W2045764924","https://openalex.org/W2051571257","https://openalex.org/W2058540481","https://openalex.org/W2058730231","https://openalex.org/W2059800803","https://openalex.org/W2073001576","https://openalex.org/W2073185604","https://openalex.org/W2074823405","https://openalex.org/W2078368926","https://openalex.org/W2078805352","https://openalex.org/W2079999798","https://openalex.org/W2085036336","https://openalex.org/W2093510572","https://openalex.org/W2108968630","https://openalex.org/W2109750460","https://openalex.org/W2127377420","https://openalex.org/W2128849971","https://openalex.org/W2138034402","https://openalex.org/W2147202919","https://openalex.org/W2163552400","https://openalex.org/W2165251681","https://openalex.org/W2167342173","https://openalex.org/W2169772316","https://openalex.org/W2275376313","https://openalex.org/W2291208220","https://openalex.org/W2296378459","https://openalex.org/W2313702768","https://openalex.org/W2341316779","https://openalex.org/W2477457213","https://openalex.org/W2487483580","https://openalex.org/W2491547471","https://openalex.org/W2491918548","https://openalex.org/W2493333586","https://openalex.org/W2499925185","https://openalex.org/W2529465479","https://openalex.org/W2685774512","https://openalex.org/W2766243828","https://openalex.org/W2802106770","https://openalex.org/W2902872320","https://openalex.org/W2911261910","https://openalex.org/W2939886985","https://openalex.org/W2971272990","https://openalex.org/W2996207370","https://openalex.org/W3031227684","https://openalex.org/W3032306896","https://openalex.org/W3035387947","https://openalex.org/W3037614249","https://openalex.org/W3043368756","https://openalex.org/W3099555404","https://openalex.org/W3101337583","https://openalex.org/W3105560263","https://openalex.org/W3122736276","https://openalex.org/W3140918176","https://openalex.org/W3151180653","https://openalex.org/W3157536011","https://openalex.org/W3173578078","https://openalex.org/W3196010733","https://openalex.org/W4205976427","https://openalex.org/W4210798852","https://openalex.org/W4213290992","https://openalex.org/W4229738979","https://openalex.org/W4230742025","https://openalex.org/W4233046245","https://openalex.org/W4248432170","https://openalex.org/W4280645728","https://openalex.org/W4285037784","https://openalex.org/W4300466986","https://openalex.org/W4312193194","https://openalex.org/W6600061385","https://openalex.org/W6640991531","https://openalex.org/W6665193932","https://openalex.org/W6668857717","https://openalex.org/W6670576825","https://openalex.org/W6722333046","https://openalex.org/W6732651597","https://openalex.org/W6761207363","https://openalex.org/W6792947318","https://openalex.org/W6806013012"],"related_works":["https://openalex.org/W4391375266","https://openalex.org/W1979597421","https://openalex.org/W2007980826","https://openalex.org/W2061531152","https://openalex.org/W3002753104","https://openalex.org/W2077600819","https://openalex.org/W4299610791","https://openalex.org/W2950866767","https://openalex.org/W2142036596","https://openalex.org/W1631776469"],"abstract_inverted_index":{"The":[0,261],"existence":[1],"of":[2,18,31,59,128,136,143,177,196,243,280,288],"a":[3,7,81,87,98,111,140,160,174,190,205,238],"maximal":[4,88,171],"ideal":[5,89,130],"in":[6,42,74,90,210,251,257],"general":[8],"nontrivial":[9],"commutative":[10,289],"ring":[11,69,96],"is":[12,70,78,97,104,116,227,237],"tied":[13],"together":[14],"with":[15,46,107,162,212,269,277],"the":[16,28,63,68,91,94,125,167,178,194,197,201,217,244,267,286,294],"axiom":[17],"choice.":[19],"Following":[20],"Berardi,":[21],"Valentini":[22],"and":[23,61,106,149,158,215,240,249,283,299],"thus":[24],"Krivine":[25],"but":[26],"using":[27],"relative":[29],"interpretation":[30],"negation":[32,181],"(that":[33],"is,":[34],"as":[35,173],"\u201cimplies":[36],"0":[37],"=":[38],"1\u201d)":[39],"we":[40,85,151,203],"show,":[41],"constructive":[43],"set":[44],"theory":[45],"minimal":[47,213],"logic,":[48,214],"how":[49],"for":[50,124,208],"countable":[51],"rings":[52,135,157],"one":[53],"can":[54],"do":[55,204],"without":[56,62],"any":[57],"kind":[58],"choice":[60],"usual":[64],"decidability":[65],"assumption":[66],"that":[67,93],"strongly":[71],"discrete":[72],"(membership":[73],"finitely":[75],"generated":[76],"ideals":[77,172],"decidable).":[79],"By":[80,138],"functional":[82],"recursive":[83],"definition":[84],"obtain":[86],"sense":[92],"quotient":[95],"residue":[99],"field":[100,113],"(every":[101,114],"noninvertible":[102],"element":[103,115],"zero),":[105],"strong":[108],"discreteness":[109],"even":[110],"geometric":[112],"either":[117],"invertible":[118],"or":[119],"else":[120],"zero).":[121],"Krull\u2019s":[122],"lemma":[123],"related":[126],"notion":[127],"prime":[129],"follows":[131],"by":[132],"passing":[133,292],"to":[134,147,155,170,187,219,285,296],"fractions.":[137],"employing":[139],"construction":[141,218],"variant":[142],"set-theoretic":[144,281],"forcing":[145],"due":[146],"Joyal":[148],"Tierney,":[150],"expand":[152],"our":[153],"treatment":[154],"arbitrary":[156,220],"establish":[159],"connection":[161,184,268,276],"dynamical":[163,168,198,270],"algebra:":[164],"We":[165],"recover":[166],"approach":[169],"parametrized":[175],"version":[176,242],"celebrated":[179],"double":[180],"translation.":[182],"This":[183,235],"allows":[185],"us":[186],"give":[188],"formal":[189],"priori":[191],"criteria":[192],"elucidating":[193],"scope":[195],"method.":[199],"Along":[200],"way":[202],"case":[206,287],"study":[207],"proofs":[209],"algebra":[211,290],"generalize":[216],"inconsistency":[221,297],"predicates.":[222],"A":[223],"partial":[224],"Agda":[225],"formalization":[226],"available":[228],"at":[229],"an":[230],"accompanying":[231],"repository.11":[232],"See":[233],"https://github.com/iblech/constructive-maximal-ideals/.":[234],"text":[236],"revised":[239],"extended":[241],"conference":[245,262],"paper":[246,263],"(In":[247],"Revolutions":[248],"Revelations":[250],"Computability.":[252],"18th":[253],"Conference":[254],"on":[255,293],"Computability":[256],"Europe":[258],"(2022)":[259],"Springer).":[260],"only":[264],"briefly":[265],"sketched":[266],"algebra;":[271],"did":[272],"not":[273],"compare":[274],"this":[275],"other":[278],"flavors":[279],"forcing;":[282],"sticked":[284],"only,":[291],"generalization":[295],"predicates":[298],"well-orders.":[300]},"counts_by_year":[],"updated_date":"2026-04-24T08:23:43.765630","created_date":"2025-10-10T00:00:00"}