An element a of a commutative ring R
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.