A nilregular element property |
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Authors: | Thierry Coquand Henri Lombardi Peter Schuster |
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Institution: | (1) University of Göteborg, Chalmers, Sweden;(2) Equipe de Mathématiques, CNRS UMR 6623, UFR des Sciences et Techniques, Université de Franche-Comté, F-25 030 Besancon cedex, France;(3) Mathematisches Institut, Universität München, Theresienstrasse 39, D-80333 München, Germany |
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Abstract: | An element a of a commutative ring R is nilregular if and only if x is nilpotent whenever ax is nilpotent. More generally, an ideal I of R is nilregular if and only if x is nilpotent whenever ax is nilpotent for all a ∈ I . We give a direct proof that if R 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 R 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.Received: 6 September 2004 |
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Keywords: | 03F65 (14M10) |
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