On Reductions of Finitely Generated Ideals in Integral Domains

√(f1,...,fd+1)R[X] [3,p.124]. The question is whether an ideal (f1,...,fd+1) R[X] can be chosen as a reduction of I. We only know the following case of affine domains, which was developed by G. Lyubeznik [4]: Let R be an n-dimensional affine domain over an infinite field k and let I be an ideal of R. Then I has a reduction generated by n+1 elements. He also posed the following conjecture: Let A be a Noetherian ring of dimension n -1 such that the residue field of every maximal ideal of A is infinite. Let I be an ideal of A or A[X] (a polynomial ring. Then I has a reduction generated by n elements. Our objective of this paper is to prove Lyubeznik's conjecture for a Noetherian domain containing an algebraically closed field: Let A be a Noetherian domain containing an algebraically closed field k and let I be an ideal of a polynomial ring A[X] such that I contains a monic