On a question of Kaplansky.

Kaplansky [7] proved that C K ( 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 C K ( 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. [ABSTRACT FROM AUTHOR]