Constructing Rings of Continuous Functions in Which There are Many Maximal Ideals With Nontrivial Rank
by
Suzanne Larson
Loyola Marymount University

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 in X, we define the rank of x to be the number of minimal prime ideals contained in the maximal ideal M_x = {f in 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). Compact SV-spaces have the property that every point has finite rank. It is an open question as to whether or not, for compact spaces, the converse holds. Two related questions appearing in the literature are 1) Is every compact SV-space a union of finitely many compact F-spaces? and 2) In a compact space where every point has finite rank, must the set of points of rank one be open? We will show how to construct spaces with many points of rank n, for any constant n, and provide a negative answer to both questions.

Date received: January 18, 2002