Pseudo-valuation rings and C(X)

A well-known line of study in the theory of the ring C(X) of continuous real-valued functions on the topological space X is to determine when, for a prime P∈Spec(C(X)), the factor ring C(X)/P is a valuation domain. In the context of commutative rings, the notion of a pseudo-valuation domain plays a fascinating role, and we investigate the question of when C(X)/P is a pseudo-valuation domain for compact Hausdorff space X and, more generally, when A is a pseudo-valuation domain for bounded real closed domain A. In this context, we note that C(X)/P is always a divided domain and show that it may be a pseudo-valuation domain without being a valuation domain.