O. IGHEDO, G. W. KIVUNGA, AND D. N. STEPHEN

Abstract

Let RL denote the ring of real-valued continuous functions on a completely regular frame L, βLand λLdenote the Stone-C^ˇech compactifica- tion of L and the Lindel¨of coreflection of L, respectively. There is a natural way of associating with each sublocale of βLtwo ideals of RL, motivated by a similar situation in C(X). This research augments the work of T. Dube and Stephen D.N. on mapping ideals to sublocales, where they associate with each sublocale of λLan ideal of RLin a manner similar to one of the ways one does it for sublocales of βL. Two other coreflections; namely, the realcompact and the paracompact coreflections are considered.

In the talk, I will 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 M_Sfor some sublocale S of L. A similar definition is given for an O-ideal of RL. Time allowing, I will present charac- terizations of M-ideals of RL indexed by spatial sublocales of βL, and O-ideals of RLindexed by closed sublocales of βLin terms of grounded maximal ideals of RL.