A nilregular element property.

Paper in proceeding, 2005

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 el ement. In constructive mathematics, this proof can then be seen as an algorithm to produce nilregular elements of nilegular 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 Gröbner basis. We dipense with prime ideals when defining concepts or carying out proofs