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A Nilregular Element Property

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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.

BibTeX - Entry

@InProceedings{coquand_et_al:DagSemProc.05021.4,
  author =	{Coquand, Thierry and Lombardi, Henri and Schuster, Peter},
  title =	{{A Nilregular Element Property}},
  booktitle =	{Mathematics, Algorithms, Proofs},
  pages =	{1--6},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{5021},
  editor =	{Thierry Coquand and Henri Lombardi and Marie-Fran\c{c}oise Roy},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2006/278},
  URN =		{urn:nbn:de:0030-drops-2784},
  doi =		{10.4230/DagSemProc.05021.4},
  annote =	{Keywords: Lists of generators, polynomial ideals, Krull dimension, Zariski topology, commutative Noetherian rings, constructive algebra}
}

Keywords: Lists of generators, polynomial ideals, Krull dimension, Zariski topology, commutative Noetherian rings, constructive algebra
Seminar: 05021 - Mathematics, Algorithms, Proofs
Issue date: 2006
Date of publication: 16.01.2006


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