A little more on ideals associated with sublocales

As usual, let RL denote the ring of real-valued continuous functions on a completely regular frame L. Let beta L and lambda L denote the Stone-Cech compactification of L and the Lindelof coreflection of L, respectively. There is a natural way of associating with each sublocale of beta L 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 lambda L an ideal of RL in a manner similar to one of the ways one does it for sublocales of beta L. 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 beta L 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 beta L, and O-ideals of RL indexed by closed sublocales of beta L in terms of grounded maximal ideals of RL.