A universal algorithm for Krull's theorem

We give a computational interpretation to an abstract formulation of Krull's theorem, by analysing its classical proof based on Zorn's lemma. Our approach is inspired by proof theory, and uses a form of update recursion to replace the existence of maximal ideals. Our main result allows us...
Ausführliche Beschreibung

Gespeichert in:

Autor*in:	Powell, Thomas [verfasserIn] Schuster, Peter [verfasserIn] Wiesnet, Franziskus [verfasserIn]

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2021

Schlagwörter:	Krull's theorem Maximal ideals Program extraction Constructive algebra

Übergeordnetes Werk:	Enthalten in: Information and computation - Amsterdam : Elsevier, 1987, 287
Übergeordnetes Werk:	volume:287

DOI / URN:	10.1016/j.ic.2021.104761

Katalog-ID:	ELV008027900

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245	1	0	\|a A universal algorithm for Krull's theorem
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520			\|a We give a computational interpretation to an abstract formulation of Krull's theorem, by analysing its classical proof based on Zorn's lemma. Our approach is inspired by proof theory, and uses a form of update recursion to replace the existence of maximal ideals. Our main result allows us to derive, in a uniform way, algorithms which compute witnesses for existential theorems in countable abstract algebra. We give a number of concrete examples of this phenomenon, including the prime ideal theorem and Krull's theorem on valuation rings.
650		4	\|a Krull's theorem
650		4	\|a Maximal ideals
650		4	\|a Program extraction
650		4	\|a Constructive algebra
700	1		\|a Schuster, Peter \|e verfasserin \|4 aut
700	1		\|a Wiesnet, Franziskus \|e verfasserin \|4 aut
773	0	8	\|i Enthalten in \|t Information and computation \|d Amsterdam : Elsevier, 1987 \|g 287 \|h Online-Ressource \|w (DE-627)26688170X \|w (DE-600)1468010-5 \|w (DE-576)10337308X \|x 1090-2651 \|7 nnns
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912			\|a GBV_ILN_95
912			\|a GBV_ILN_100
912			\|a GBV_ILN_101
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912			\|a GBV_ILN_110
912			\|a GBV_ILN_150
912			\|a GBV_ILN_151
912			\|a GBV_ILN_224
912			\|a GBV_ILN_370
912			\|a GBV_ILN_602
912			\|a GBV_ILN_702
912			\|a GBV_ILN_2003
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912			\|a GBV_ILN_2190
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912			\|a GBV_ILN_2336
912			\|a GBV_ILN_2470
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allfields	10.1016/j.ic.2021.104761 doi (DE-627)ELV008027900 (ELSEVIER)S0890-5401(21)00076-6 DE-627 ger DE-627 rda eng 330 004 DE-600 LING DE-30 fid 54.00 bkl 31.80 bkl Powell, Thomas verfasserin aut A universal algorithm for Krull's theorem 2021 nicht spezifiziert zzz rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier We give a computational interpretation to an abstract formulation of Krull's theorem, by analysing its classical proof based on Zorn's lemma. Our approach is inspired by proof theory, and uses a form of update recursion to replace the existence of maximal ideals. Our main result allows us to derive, in a uniform way, algorithms which compute witnesses for existential theorems in countable abstract algebra. We give a number of concrete examples of this phenomenon, including the prime ideal theorem and Krull's theorem on valuation rings. Krull's theorem Maximal ideals Program extraction Constructive algebra Schuster, Peter verfasserin aut Wiesnet, Franziskus verfasserin aut Enthalten in Information and computation Amsterdam : Elsevier, 1987 287 Online-Ressource (DE-627)26688170X (DE-600)1468010-5 (DE-576)10337308X 1090-2651 nnns volume:287 GBV_USEFLAG_U SYSFLAG_U GBV_ELV FID-LING SSG-OPC-MAT GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_150 GBV_ILN_151 GBV_ILN_224 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2034 GBV_ILN_2038 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2470 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 54.00 Informatik: Allgemeines 31.80 Angewandte Mathematik AR 287
spelling	10.1016/j.ic.2021.104761 doi (DE-627)ELV008027900 (ELSEVIER)S0890-5401(21)00076-6 DE-627 ger DE-627 rda eng 330 004 DE-600 LING DE-30 fid 54.00 bkl 31.80 bkl Powell, Thomas verfasserin aut A universal algorithm for Krull's theorem 2021 nicht spezifiziert zzz rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier We give a computational interpretation to an abstract formulation of Krull's theorem, by analysing its classical proof based on Zorn's lemma. Our approach is inspired by proof theory, and uses a form of update recursion to replace the existence of maximal ideals. Our main result allows us to derive, in a uniform way, algorithms which compute witnesses for existential theorems in countable abstract algebra. We give a number of concrete examples of this phenomenon, including the prime ideal theorem and Krull's theorem on valuation rings. Krull's theorem Maximal ideals Program extraction Constructive algebra Schuster, Peter verfasserin aut Wiesnet, Franziskus verfasserin aut Enthalten in Information and computation Amsterdam : Elsevier, 1987 287 Online-Ressource (DE-627)26688170X (DE-600)1468010-5 (DE-576)10337308X 1090-2651 nnns volume:287 GBV_USEFLAG_U SYSFLAG_U GBV_ELV FID-LING SSG-OPC-MAT GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_150 GBV_ILN_151 GBV_ILN_224 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2034 GBV_ILN_2038 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2470 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 54.00 Informatik: Allgemeines 31.80 Angewandte Mathematik AR 287
allfields_unstemmed	10.1016/j.ic.2021.104761 doi (DE-627)ELV008027900 (ELSEVIER)S0890-5401(21)00076-6 DE-627 ger DE-627 rda eng 330 004 DE-600 LING DE-30 fid 54.00 bkl 31.80 bkl Powell, Thomas verfasserin aut A universal algorithm for Krull's theorem 2021 nicht spezifiziert zzz rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier We give a computational interpretation to an abstract formulation of Krull's theorem, by analysing its classical proof based on Zorn's lemma. Our approach is inspired by proof theory, and uses a form of update recursion to replace the existence of maximal ideals. Our main result allows us to derive, in a uniform way, algorithms which compute witnesses for existential theorems in countable abstract algebra. We give a number of concrete examples of this phenomenon, including the prime ideal theorem and Krull's theorem on valuation rings. Krull's theorem Maximal ideals Program extraction Constructive algebra Schuster, Peter verfasserin aut Wiesnet, Franziskus verfasserin aut Enthalten in Information and computation Amsterdam : Elsevier, 1987 287 Online-Ressource (DE-627)26688170X (DE-600)1468010-5 (DE-576)10337308X 1090-2651 nnns volume:287 GBV_USEFLAG_U SYSFLAG_U GBV_ELV FID-LING SSG-OPC-MAT GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_150 GBV_ILN_151 GBV_ILN_224 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2034 GBV_ILN_2038 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2470 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 54.00 Informatik: Allgemeines 31.80 Angewandte Mathematik AR 287
allfieldsGer	10.1016/j.ic.2021.104761 doi (DE-627)ELV008027900 (ELSEVIER)S0890-5401(21)00076-6 DE-627 ger DE-627 rda eng 330 004 DE-600 LING DE-30 fid 54.00 bkl 31.80 bkl Powell, Thomas verfasserin aut A universal algorithm for Krull's theorem 2021 nicht spezifiziert zzz rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier We give a computational interpretation to an abstract formulation of Krull's theorem, by analysing its classical proof based on Zorn's lemma. Our approach is inspired by proof theory, and uses a form of update recursion to replace the existence of maximal ideals. Our main result allows us to derive, in a uniform way, algorithms which compute witnesses for existential theorems in countable abstract algebra. We give a number of concrete examples of this phenomenon, including the prime ideal theorem and Krull's theorem on valuation rings. Krull's theorem Maximal ideals Program extraction Constructive algebra Schuster, Peter verfasserin aut Wiesnet, Franziskus verfasserin aut Enthalten in Information and computation Amsterdam : Elsevier, 1987 287 Online-Ressource (DE-627)26688170X (DE-600)1468010-5 (DE-576)10337308X 1090-2651 nnns volume:287 GBV_USEFLAG_U SYSFLAG_U GBV_ELV FID-LING SSG-OPC-MAT GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_150 GBV_ILN_151 GBV_ILN_224 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2034 GBV_ILN_2038 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2470 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 54.00 Informatik: Allgemeines 31.80 Angewandte Mathematik AR 287
allfieldsSound	10.1016/j.ic.2021.104761 doi (DE-627)ELV008027900 (ELSEVIER)S0890-5401(21)00076-6 DE-627 ger DE-627 rda eng 330 004 DE-600 LING DE-30 fid 54.00 bkl 31.80 bkl Powell, Thomas verfasserin aut A universal algorithm for Krull's theorem 2021 nicht spezifiziert zzz rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier We give a computational interpretation to an abstract formulation of Krull's theorem, by analysing its classical proof based on Zorn's lemma. Our approach is inspired by proof theory, and uses a form of update recursion to replace the existence of maximal ideals. Our main result allows us to derive, in a uniform way, algorithms which compute witnesses for existential theorems in countable abstract algebra. We give a number of concrete examples of this phenomenon, including the prime ideal theorem and Krull's theorem on valuation rings. Krull's theorem Maximal ideals Program extraction Constructive algebra Schuster, Peter verfasserin aut Wiesnet, Franziskus verfasserin aut Enthalten in Information and computation Amsterdam : Elsevier, 1987 287 Online-Ressource (DE-627)26688170X (DE-600)1468010-5 (DE-576)10337308X 1090-2651 nnns volume:287 GBV_USEFLAG_U SYSFLAG_U GBV_ELV FID-LING SSG-OPC-MAT GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_150 GBV_ILN_151 GBV_ILN_224 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2034 GBV_ILN_2038 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2470 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 54.00 Informatik: Allgemeines 31.80 Angewandte Mathematik AR 287
language	English
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sourceStr	Enthalten in Information and computation 287 volume:287
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topic_facet	Krull's theorem Maximal ideals Program extraction Constructive algebra
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container_title	Information and computation
authorswithroles_txt_mv	Powell, Thomas @@aut@@ Schuster, Peter @@aut@@ Wiesnet, Franziskus @@aut@@
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author	Powell, Thomas
spellingShingle	Powell, Thomas ddc 330 fid LING bkl 54.00 bkl 31.80 misc Krull's theorem misc Maximal ideals misc Program extraction misc Constructive algebra A universal algorithm for Krull's theorem
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topic_title	330 004 DE-600 LING DE-30 fid 54.00 bkl 31.80 bkl A universal algorithm for Krull's theorem Krull's theorem Maximal ideals Program extraction Constructive algebra
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title	A universal algorithm for Krull's theorem
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title_full	A universal algorithm for Krull's theorem
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title_sort	a universal algorithm for krull's theorem
title_auth	A universal algorithm for Krull's theorem
abstract	We give a computational interpretation to an abstract formulation of Krull's theorem, by analysing its classical proof based on Zorn's lemma. Our approach is inspired by proof theory, and uses a form of update recursion to replace the existence of maximal ideals. Our main result allows us to derive, in a uniform way, algorithms which compute witnesses for existential theorems in countable abstract algebra. We give a number of concrete examples of this phenomenon, including the prime ideal theorem and Krull's theorem on valuation rings.
abstractGer	We give a computational interpretation to an abstract formulation of Krull's theorem, by analysing its classical proof based on Zorn's lemma. Our approach is inspired by proof theory, and uses a form of update recursion to replace the existence of maximal ideals. Our main result allows us to derive, in a uniform way, algorithms which compute witnesses for existential theorems in countable abstract algebra. We give a number of concrete examples of this phenomenon, including the prime ideal theorem and Krull's theorem on valuation rings.
abstract_unstemmed	We give a computational interpretation to an abstract formulation of Krull's theorem, by analysing its classical proof based on Zorn's lemma. Our approach is inspired by proof theory, and uses a form of update recursion to replace the existence of maximal ideals. Our main result allows us to derive, in a uniform way, algorithms which compute witnesses for existential theorems in countable abstract algebra. We give a number of concrete examples of this phenomenon, including the prime ideal theorem and Krull's theorem on valuation rings.
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title_short	A universal algorithm for Krull's theorem
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up_date	2024-07-06T18:17:28.863Z
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A universal algorithm for Krull's theorem

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