A Nilregular Element Property

Authors Thierry Coquand, Henri Lombardi, Peter Schuster

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Thierry Coquand
Henri Lombardi
Peter Schuster

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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)


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.
  • Lists of generators
  • polynomial ideals
  • Krull dimension
  • Zariski topology
  • commutative Noetherian rings
  • constructive algebra


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