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.
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Received: 6 September 2004
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Coquand, T., Lombardi, H. & Schuster, P. A nilregular element property. Arch. Math. 85, 49–54 (2005). https://doi.org/10.1007/s00013-005-1295-0
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DOI: https://doi.org/10.1007/s00013-005-1295-0