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

Thomas Powell, Peter Schuster, Franziskus Wiesnet

Research output: Chapter in Book/Report/Conference proceedingConference contribution

3 Citations (Scopus)

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 languageEnglish
Title of host publicationLogic, Language, Information, and Computation
Subtitle of host publication26th International Workshop, WoLLIC 2019, Utrecht, The Netherlands, July 2-5, 2019, Proceedings
Pages533-549
DOIs
Publication statusPublished - 2019

Publication series

NameLecture Notes in Computer Science
PublisherSpringer
Volume11541
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Cite this

Powell, T., Schuster, P., & Wiesnet, F. (2019). An algorithmic approach to the existence of ideal objects in commutative algebra. In 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; Vol. 11541). https://doi.org/10.1007/978-3-662-59533-6_32