Rings of quotients of rings of functions

If X is a Tychonoff space, a zero-set Z of X is z-complemented in X if there exists a zero-set Z of X such that Z∪Z=X and Z∩Z is nowhere dense in X. The notion of z-complemented zero-sets arises in determining the rings of continuous functions C(X) having the property that the total ring of quotients T(C(X)) is von Neumann regular. In this note, we first examine conditions on a space X under which every zero-set is z-complemented. Then in Section 4 we relate z-Gabriel filters in the ring C(X) to certain filters of open sets of X and in some instances we show how the localization of C(X) at such a filter is isomorphic to a ring of partial functions on a subspace of X.