Goldie dimension of rings of fractions of C(X)

Abstract

It is observed that X is an F-space if and only if C(X) is locally a domain (i.e., C(X)_P is a domain for each prime ideal P of C(X)). Consequently, X is an F-space if and only if the primary ideals of C(X) in any given maximal ideal in C(X) are comparable. Some of the properties of C(X), where X is an F-space, are extended to general reduced Bézout rings. It is observed that whenever X is an infinite connected F-space, then C(X) is a natural example of a non-Noetherian ring without nontrivial idempotents which is locally a domain but not a domain. We observe that the rank of a point x ∈ βX, in case finite, coincides with the Goldie dimension of C(X)_M^x and give an example to show that the Goldie dimension of C(X)_M^x is not necessarily equal to the cardinality of the set of minimal prime ideals in M^x. Motivated by these facts and some other appropriate ones, we define the rank of a point x ∈ βX to be the Goldie dimension of C(X)_M^x. Finally, for each cardinal a, we show that there exists a space X and a multiplicatively closed set S in C(X) such that the Goldie dimension of S^–1 C(X) is a.