Coquand, Thierry ;
Lombardi, Henri ;
Schuster, Peter
A Nilregular Element Property
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 EisenbudEvansStorch 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:DSP:2006:278,
author = {Thierry Coquand and Henri Lombardi and Peter Schuster},
title = {A Nilregular Element Property},
booktitle = {Mathematics, Algorithms, Proofs},
year = {2006},
editor = {Thierry Coquand and Henri Lombardi and MarieFran{\c{c}}oise Roy},
number = {05021},
series = {Dagstuhl Seminar Proceedings},
ISSN = {18624405},
publisher = {Internationales Begegnungs und Forschungszentrum f{\"u}r Informatik (IBFI), Schloss Dagstuhl, Germany},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2006/278},
annote = {Keywords: Lists of generators, polynomial ideals, Krull dimension, Zariski topology, commutative Noetherian rings, constructive algebra}
}
2006
Keywords: 

Lists of generators, polynomial ideals, Krull dimension, Zariski topology, commutative Noetherian rings, constructive algebra 
Seminar: 

05021  Mathematics, Algorithms, Proofs

Related Scholarly Article: 


Issue date: 

2006 
Date of publication: 

2006 