A nilregular element property.

Thierry Coquand, Henri Lombardi and Peter Schuster.
Archiv der Mathematik. 85 (2005), 49-54.

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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 in I.
We give a direct proof that if R is noetherian, then any nilregular ideal contains a nilregular element. In constructive mathematics, this proof can then be seen as an algorithm to produce nilregular elements whenever R is coherent, noetherian, and strongly 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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