2026-08-10T13:32:00Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:278 2024-03-06T11:06:07Z ddc:004 open_access

A Nilregular Element Property Coquand, Thierry Lombardi, Henri Schuster, Peter Lists of generators polynomial ideals Krull dimension Zariski topology commutative Noetherian rings constructive algebra 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Thierry Coquand and Henri Lombardi and Peter Schuster 2006 Is Part Of Dagstuhl Seminar Proceedings, Volume 5021, Mathematics, Algorithms, Proofs (2006) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/DagSemProc.05021.4 urn:nbn:de:0030-drops-2784 https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.05021.4 eng https://creativecommons.org/licenses/by/4.0/legalcode