Constructing Rings of Continuous Functions in Which There Are Many Maximal Ideals with Nontrivial Rank

Abstract

Let X be a topological space, and let C (X) denote the f-ring of all continuous real-valued functions defined on X. For x ∈ X, we define the rank of x to be the number of minimal prime ideals contained in the maximal ideal M x = {f ∈ C (X) : f(x) = 0} if there are finitely many such minimal prime ideals, and the rank of x to be infinite if there are infinitely many minimal prime ideals contained in M x . We call X an SV-space if C (X)/P is a valuation domain for each minimal prime ideal P of C (X). A construction of topological spaces is given showing that a compact SV-space need not be a union of finitely many compact F-spaces and that in a compact space where every point has finite rank, the set of points of rank one need not be open.