#### Title

When Is C(X)/P a Valuation Ring for Every Prime Ideal P?

#### Document Type

Article

#### Department

Mathematics (HMC)

#### Publication Date

5-22-1992

#### Abstract

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 υ*X* 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.

#### Rights Information

© 1992 Elsevier

#### DOI

10.1016/0166-8641(92)90091-D

#### Recommended Citation

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.