Prime Ideals Lying over Zero in Polynomial Rings Original Research Article

Let R be a Noetherian domain and x an indeterminate. Let image and image be two finite sets of prime ideals in the polynomial ring R[x]. Necessary and sufficient conditions are given for the existence of a prime ideal K in R[x] such that K ∩ R = O, K contains a monic polynomial, and K is contained in every prime ideal in image but not in any prime ideal in image.