On a question of Kaplansky

Kaplansky [7] proved that CK(X) is the intersection of all free maximal ideals in C(X) in the case of discrete X, and asked whether the equality holds in general. In this paper we prove that CK(X) coincides with the intersection of all free maximal ideals if and only if every open hemicompact z-compact (i.e., every zero-set contained in it is compact) subset of X is relatively compact or equivalently, every open Lindelöf z-compact subset of X is relatively compact. We conclude that the equality holds whenever X is a strongly isocompact space.