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

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.