When Is C(X)/P a Valuation Ring for Every Prime Ideal P?
A Tychonoff space X is called an SV-space if for every prime ideal P of the ring C(X) of continuous real-valued functions on X, the ordered integral domain C(X)/P is a valuation ring (i.e., of any two nonzero elements of C(X)/P, one divides the other). It is shown that X is an SV-space iff υX is an SV-space iff βX is an SV-space. If every point of X has a neighborhood that is an F-space, then X is an SV-space. An example is supplied of an infinite compact SV-space such that any point with an F-space neighborhood is isolated. It is shown that the class of SV-spaces includes those Tychonoff spaces that are finite unions of C*-embedded SV-spaces. Some open problems are posed.
Henriksen, Melvin and Wilson, Richard. 1991. When is C(X)/P a valuation ring for every prime ideal P? Topology and its Applications. 44(1-3):175-180. < http://www.sciencedirect.com/science/article/B6V1K-45FCWBM-16/2/cf0035dadd7ea6ad1f322178e0a77d3e>.