DagSemProc.05021.4.pdf
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A Nilregular Element Property
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Dagstuhl Seminar Proceedings, Volume 5021
Series: Dagstuhl Seminar Proceedings (DagSemProc) - License:
Creative Commons Attribution 4.0 International license
- Publication Date: 2006-01-16
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Thierry Coquand, Henri Lombardi, and Peter Schuster. A Nilregular Element Property. In Mathematics, Algorithms, Proofs. Dagstuhl Seminar Proceedings, Volume 5021, pp. 1-6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2006)
https://doi.org/10.4230/DagSemProc.05021.4
Abstract
An element or an ideal of a commutative ring is nilregular if and only if it is regular modulo the nilradical. We prove that if the ring is Noetherian, then every nilregular ideal contains a nilregular element. In constructive mathematics, this proof can then be seen as an algorithm to produce nilregular elements of nilregular ideals whenever the ring is coherent, Noetherian, and discrete. As an application, we give a constructive proof of the Eisenbud--Evans--Storch theorem that every algebraic set in $n$--dimensional affine space is the intersection of $n$ hypersurfaces. The input of the algorithm is an arbitrary finite list of polynomials, which need not arrive in a special form such as a Gr"obner basis. We dispense with prime ideals when defining concepts or carrying out proofs.
Subject Classification
Keywords
- Lists of generators
- polynomial ideals
- Krull dimension
- Zariski topology
- commutative Noetherian rings
- constructive algebra
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