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As usual, let RL denote the ring of real-valued continuous functions on a completely regular frame L. Let βL and λL denote the Stone- Čech compactification of L and the Lindelöf 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. AnM-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.


This article was originally published in Categories and General Algebraic Structures with Applications , volume 20, issue 1, in 2024.

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Shahid Beheshti University

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