A little more on ideals associated with sublocales

As usual, let \mathcal RL denote the ring of real-valued continuous functions on a completely regular frame L. Let \beta L and  \lambda L denote the  Stone-\v{C}ech compactification of L and the Lindel\"of coreflection of L, respectively. There is a natural way of associating with each sublocale of \beta L two ideals of \mathcal RL, motivated by a similar situation in C(X). In~\cite{DS۱}, the authors go one step further and associate with each sublocale  of \lambda L an ideal of \mathcal 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~\cite{DS۱} by considering two other coreflections; namely, the realcompact and the paracompact   coreflections.\\        We show that \boldsymbol M-ideals of \mathcal RL indexed by sublocales of \beta L are precisely the intersections of maximal ideals of  \mathcal RL. An \boldsymbol{M}-ideal of \mathcal RL is \emph{grounded} in case it is of the form \boldsymbol{M}_S for some sublocale S of L. A similar definition is given for an  \boldsymbol{O}-ideal of \mathcal RL.  We characterise \boldsymbol M-ideals of \mathcal RL indexed by spatial sublocales of \beta L, and \boldsymbol O-ideals of \mathcal RL indexed by closed sublocales of \beta L in terms of grounded maximal ideals of \mathcal RL.

Keywords:

frame , locale , sublocale , pointfree function ring , Lindel\"{o}f , realcompact , paracompact