Hilbert integral domains with maximal ideals of preassigned height

Abstract

A commutative ring with identity R is called a Hilbert ring if every prime ideal in R is an intersection of maximal ideals. It is well known that, if R is a Hilbert ring, then the polynomial ring R[X] is also a Hilbert ring. Indeed, Hilbert rings first arose in this connection: the fact that R is Hilbert implies R[X] is Hilbert, being used by Krull in [4] and Goldman in [l] to give a ring theoretic proof for the Hilbert Nullstellensatz. If R is a Hilbert ring, then every maximal ideal in R[X] lies over a maximal ideal in R (see, for example, [3, p. 181). Thus, if R is a Hilbert ring and every maximal ideal in R has height >n, then every maximal ideal in R[XJ has height >n + 1. (A prime ideal P in R is said to have height n if there is a chain P,, < P1 < -*a < P, = P of prime ideals in R but no such chain of longer length.) The most naturally occurring examples of Hilbert domains are finitely generated ring extensions of a field or the ring of integers. Such examples as these have the property that all maximal ideals have the same height. Leslie Roberts in [5] points out that Hilbert domains with maximal ideals of different height do not seem to be well known, and gives, by means of an interesting localization technique, examples of Hilbert domains having maximal ideals of height both n and n + 1. The purpose of this note is to construct Hilbert domains with maximal ideals of various preassigned height. This is done by making use of intersection properties for certain integral domains given by Jack Ohm and the author in [2].