# Ways in which C(X) mod a Prime Ideal Can be a Valuation Domain; Something Old and Something New

## Document Type

Book Chapter

## Department

Mathematics (HMC)

## Publication Date

2007

## Abstract

*C*(*X*) denotes the ring of continuous real-valued functions on a Tychonoff space *X* and *P* a prime ideal of *C*(*X*). We summarize a lot of what is known about the reside class domains *C*(*X*)/*P* and add many new results about this subject with an emphasis on determining when the ordered *C*(*X*)/*P* is a valuation domain (i.e., when given two nonzero elements, one of them must divide the other). The interaction between the space *X* and the prime ideal *P* is of great importance in this study. We summarize first what is known when *P* is a maximal ideal, and then what happens when *C*(*X*)/*P* is a valuation domain for every prime ideal *P* (in which case *X* is called an *SV*-space and *C*(*X*) an *SV*-ring). Two new generalizations are introduced and studied. The first is that of an almost *SV*-spaces in which each maximal ideal contains a minimal prime ideal *P* such that *C*(*X*)/*P* is a valuation domain. In the second, we assume that each real maximal ideal that fails to be minimal contains a nonmaximal prime ideal *P* such that *C*(*X*)/*P* is a valuation domain. Some of our results depend on whether or not *βω ω* contains a *P*-point. Some concluding remarks include unsolved problems.

## Rights Information

© 2007 Springer

## DOI

10.1007/978-3-7643-8478-4_1

## Recommended Citation

Bikram Banerjee; Henriksen, Melvin. Ways in which C(X) mod a prime ideal can be a valuation domain: something old and something new. Positivity, Trends in Mathematics, Birkhauser Verlag (2007), 1–25. DOI: 10.1007/978-3-7643-8478-4_1