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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.

Recommended Citation

Suzanne Larson (2003) Constructing Rings of Continuous Functions in Which There Are Many Maximal Ideals with Nontrivial Rank, Communications in Algebra, 31:5, 2183-2206, DOI: 10.1081/AGB-120018991