Document Type
Article
Publication Date
1-2024
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 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.
Recommended Citation
Ighedo, O., Kivunga, G.W. and Stephen, D.N., A little more on ideals associated with sublocales, Categories and General Algebraic Structures with Applications 20 (2024), 175-200. https://doi.org/10.48308/cgasa.2023.234093.1456
Peer Reviewed
1
Copyright
Shahid Beheshti University
Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 License.
Comments
This article was originally published in Categories and General Algebraic Structures with Applications , volume 20, issue 1, in 2024. https://doi.org/10.48308/cgasa.2023.234093.1456