An algorithmic approach to the existence of ideal objects in commutative algebra

Thomas Powell; Peter Schuster; Franziskus Wiesnet

doi:10.1007/978-3-662-59533-6_32

An algorithmic approach to the existence of ideal objects in commutative algebra

Thomas Powell, Peter Schuster, Franziskus Wiesnet

Research output: Chapter or section in a book/report/conference proceeding › Chapter in a published conference proceeding

8 Citations (SciVal)

Abstract

The existence of ideal objects, such as maximal ideals in nonzero rings, plays a crucial role in commutative algebra. These are typically justified using Zorn’s lemma, and thus pose a challenge from a computational point of view. Giving a constructive meaning to ideal objects is a problem which dates back to Hilbert’s program, and today is still a central theme in the area of dynamical algebra, which focuses on the elimination of ideal objects via syntactic methods. In this paper, we take an alternative approach based on Kreisel’s no counterexample interpretation and sequential algorithms. We first give a computational interpretation to an abstract maximality principle in the countable setting via an intuitive, state based algorithm. We then carry out a concrete case study, in which we give an algorithmic account of the result that in any commutative ring, the intersection of all prime ideals is contained in its nilradical.

Original language	English
Title of host publication	Logic, Language, Information, and Computation
Subtitle of host publication	26th International Workshop, WoLLIC 2019, Utrecht, The Netherlands, July 2-5, 2019, Proceedings
Editors	Rosalie Iemhoff, Michael Moortgat, Ruy de Queiroz
Publisher	Springer Verlag
Pages	533-549
Number of pages	17
ISBN (Print)	9783662595329
DOIs	https://doi.org/10.1007/978-3-662-59533-6_32
Publication status	Published - 9 Jun 2019
Event	26th International Workshop on Logic, Language, Information and Communication, WoLLIC 2019 - Utrecht, Netherlands Duration: 2 Jul 2019 → 5 Jul 2019

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	11541 LNCS
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	26th International Workshop on Logic, Language, Information and Communication, WoLLIC 2019
Country/Territory	Netherlands
City	Utrecht
Period	2/07/19 → 5/07/19

Bibliographical note

Publisher Copyright:
© 2019, Springer-Verlag GmbH Germany, part of Springer Nature.

Acknowledgements

The authors are grateful to the anonymous referees for their detailed comments, which led to a much improved version of the paper.

Keywords

Commutative algebra
No-counterexample interpretation
Program extraction
Proof theory

ASJC Scopus subject areas

Theoretical Computer Science
General Computer Science

Access to Document

10.1007/978-3-662-59533-6_32

Cite this

Powell, T., Schuster, P., & Wiesnet, F. (2019). An algorithmic approach to the existence of ideal objects in commutative algebra. In R. Iemhoff, M. Moortgat, & R. de Queiroz (Eds.), Logic, Language, Information, and Computation: 26th International Workshop, WoLLIC 2019, Utrecht, The Netherlands, July 2-5, 2019, Proceedings (pp. 533-549). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 11541 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-662-59533-6_32

An algorithmic approach to the existence of ideal objects in commutative algebra. / Powell, Thomas; Schuster, Peter; Wiesnet, Franziskus.
Logic, Language, Information, and Computation: 26th International Workshop, WoLLIC 2019, Utrecht, The Netherlands, July 2-5, 2019, Proceedings. ed. / Rosalie Iemhoff; Michael Moortgat; Ruy de Queiroz. Springer Verlag, 2019. p. 533-549 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 11541 LNCS).

Research output: Chapter or section in a book/report/conference proceeding › Chapter in a published conference proceeding

Powell, T, Schuster, P & Wiesnet, F 2019, An algorithmic approach to the existence of ideal objects in commutative algebra. in R Iemhoff, M Moortgat & R de Queiroz (eds), Logic, Language, Information, and Computation: 26th International Workshop, WoLLIC 2019, Utrecht, The Netherlands, July 2-5, 2019, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 11541 LNCS, Springer Verlag, pp. 533-549, 26th International Workshop on Logic, Language, Information and Communication, WoLLIC 2019, Utrecht, Netherlands, 2/07/19. https://doi.org/10.1007/978-3-662-59533-6_32

Powell T, Schuster P, Wiesnet F. An algorithmic approach to the existence of ideal objects in commutative algebra. In Iemhoff R, Moortgat M, de Queiroz R, editors, Logic, Language, Information, and Computation: 26th International Workshop, WoLLIC 2019, Utrecht, The Netherlands, July 2-5, 2019, Proceedings. Springer Verlag. 2019. p. 533-549. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/978-3-662-59533-6_32

Powell, Thomas ; Schuster, Peter ; Wiesnet, Franziskus. / An algorithmic approach to the existence of ideal objects in commutative algebra. Logic, Language, Information, and Computation: 26th International Workshop, WoLLIC 2019, Utrecht, The Netherlands, July 2-5, 2019, Proceedings. editor / Rosalie Iemhoff ; Michael Moortgat ; Ruy de Queiroz. Springer Verlag, 2019. pp. 533-549 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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An algorithmic approach to the existence of ideal objects in commutative algebra

Abstract

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Bibliographical note

Acknowledgements

Keywords

ASJC Scopus subject areas

Access to Document

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