Induction in Algebra: a First Case Study

Peter M Schuster

doi:10.2168/LMCS-9(3:20)2013

Peter M Schuster - Induction in Algebra: a First Case Study

lmcs:959 - Logical Methods in Computer Science, September 17, 2013, Volume 9, Issue 3 - https://doi.org/10.2168/LMCS-9(3:20)2013

Induction in Algebra: a First Case StudyArticle

Authors: Peter M Schuster

Many a concrete theorem of abstract algebra admits a short and elegant proof by contradiction but with Zorn's Lemma (ZL). A few of these theorems have recently turned out to follow in a direct and elementary way from the Principle of Open Induction distinguished by Raoult. The ideal objects characteristic of any invocation of ZL are eliminated, and it is made possible to pass from classical to intuitionistic logic. If the theorem has finite input data, then a finite partial order carries the required instance of induction, which thus is constructively provable. A typical example is the well-known theorem "every nonconstant coefficient of an invertible polynomial is nilpotent".

https://doi.org/10.2168/LMCS-9(3:20)2013

Source: arXiv.org:1308.2690

Volume: Volume 9, Issue 3

Published on: September 17, 2013

Imported on: March 16, 2013

Keywords: Computer Science - Logic in Computer Science, Mathematics - Commutative Algebra, Mathematics - Logic

Licence: arXiv.org - Non-exclusive license to distribute

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Mathematics Subject Classification 2020¹

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[1] zbMATH Open.

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Review available on zbMATH Open, by Johan Georg Granström (Zürich) (English)

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16 Documents citing this article

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Ingo Blechschmidt;Peter Schuster, 2022, Maximal Ideals in Countable Rings, Constructively, Lecture notes in computer science, pp. 24-38, 10.1007/978-3-031-08740-0_3.

Peter Schuster;Daniel Wessel, 2022, The Jacobson radical for an inconsistency predicate, Computability, 11, 2, pp. 147-162, 10.3233/com-210365, https://doi.org/10.3233/com-210365.

Peter Schuster;Daniel Wessel, 2021, Syntax for Semantics: Krull’s Maximal Ideal Theorem, Logic, epistemology, and the unity of science, pp. 77-102, 10.1007/978-3-030-65824-3_6, https://doi.org/10.1007/978-3-030-65824-3_6.

Peter Schuster;Daniel Wessel;Ihsen Yengui, 2021, Dynamic evaluation of integrity and the computational content of Krull's lemma, Journal of Pure and Applied Algebra, 226, 1, pp. 106794, 10.1016/j.jpaa.2021.106794.

Peter Schuster;Daniel Wessel, 2020, Resolving finite indeterminacy, Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science, pp. 820-830, 10.1145/3373718.3394777.

Peter Schuster;Daniel Wessel, 2020, The Computational Significance of Hausdorff’s Maximal Chain Principle, PubMed Central, pp. 239-250, 10.1007/978-3-030-51466-2_21, https://www.ncbi.nlm.nih.gov/pmc/articles/7309501.

Thomas Powell;Peter Schuster;Franziskus Wiesnet, 2019, An Algorithmic Approach to the Existence of Ideal Objects in Commutative Algebra, arXiv (Cornell University), pp. 533-549, 10.1007/978-3-662-59533-6_32, http://arxiv.org/abs/1903.03070.

DAVIDE RINALDI;PETER SCHUSTER;DANIEL WESSEL, 2017, ELIMINATING DISJUNCTIONS BY DISJUNCTION ELIMINATION, Bulletin of Symbolic Logic, 23, 2, pp. 181-200, 10.1017/bsl.2017.13.

Davide Rinaldi;Peter Schuster;Daniel Wessel, 2017, Eliminating disjunctions by disjunction elimination, Indagationes Mathematicae, 29, 1, pp. 226-259, 10.1016/j.indag.2017.09.011.

Davide Rinaldi;Giovanni Sambin;Peter Schuster, 2015, The Basic Zariski Topology, Confluentes Mathematici, 7, 1, pp. 55-81, 10.5802/cml.18, https://doi.org/10.5802/cml.18.

Ihsen Yengui, 2015, Exercises, Lecture notes in mathematics, pp. 207-220, 10.1007/978-3-319-19494-3_5.

Ihsen Yengui, 2015, Dynamical Gröbner Bases, Lecture notes in mathematics, pp. 105-171, 10.1007/978-3-319-19494-3_3.

Ihsen Yengui, 2015, Detailed Solutions to the Exercises, Lecture notes in mathematics, pp. 221-253, 10.1007/978-3-319-19494-3_6.

Ihsen Yengui, 2015, Introduction, Lecture notes in mathematics, pp. 1-8, 10.1007/978-3-319-19494-3_1.

Ihsen Yengui, 2015, Projective Modules Over Polynomial Rings, Lecture notes in mathematics, pp. 9-103, 10.1007/978-3-319-19494-3_2.

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