A little more on ideals associated with sublocales

Abstract

As usual, let RL denote the ring of real-valued continuous functions on a completely regular frame L. Let βL and λL denote the StoneCech compactification of ˇ L and the Lindel¨of coreflection of L, respectively. There is a natural way of associating with each sublocale of βL two ideals of RL, motivated by a similar situation in C(X). In [12], the authors go one step further and associate with each sublocale of λL an ideal of RL in a manner similar to one of the ways one does it for sublocales of βL. The intent in this paper is to augment [12] by considering two other coreflections; namely, the realcompact and the paracompact coreflections. We show that M-ideals of RL indexed by sublocales of βL are precisely the intersections of maximal ideals of RL. An M-ideal of RL is grounded in case it is of the form MS for some sublocale S of L. A similar definition is given for an O-ideal of RL. We characterise M-ideals of RL indexed by spatial sublocales of βL, and O-ideals of RL indexed by closed sublocales of βL in terms of grounded maximal ideals of RL.