Constructing rings of continuous functions in which there are many maximal ideals with nontrivial rank

摘要

Let X be a topological space, and let C(A) denote the f-ring of all continuous real-valued functions defined on X. For x E X, we define the rank of x to be the number of minimal prime ideals contained in the maximal ideal M-x= {fis an element ofC(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, 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. References: 11