Unions of minimal prime ideals in rings of continuous functions on compact spaces

Abstract

As is well-known to those familiar with the Gillman-Jerison text on the ring C(X) of real-valued continuous functions on a topological space (which we may assume is a Tychonoff space), every prime ideal is contained in a unique maximal ideal. In case X is compact, each maximal ideal is of the form Mp for some \({p \in X}\) and consists of all \({f \in C(X)}\) such that f(p) = 0, while the intersection of all minimal prime ideals contained in Mp is the set of all continuous functions which vanish on a neighborhood of p. In this paper, we reverse some inclusions and study the union of all of the minimal prime ideals contained in Mp; particularly in the case when this set-theoretic union is all of Mp. When this occurs, we call X a UMP-space. By making use of the well-known theorem of Gelfand and Kolmogoroff, we obtain new results without assuming that X is compact. It turns out that all UMP-spaces have the property that each of its nonempty zero-sets has nonempty interior. That is, X is an almost P-space. But this condition is far from sufficient.