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 Bezout rings. It is observed that whenever X is an innite 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 dene 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^-1C(X) is α.

Keywords: Goldie dimension, F-spaces, P-spaces, locally domains, reduced rings, Bezout rings, rank of a point, local cellularity, local Souslin property, at module, SV-spaces, rings of fractions.