%0 Journal Article
%T A little more on ideals associated with sublocales
%J Categories and General Algebraic Structures with Applications
%I Shahid Beheshti University
%Z 2345-5853
%A Ighedo, Oghenetega
%A Kivunga, Grace Wakesho
%A Stephen, Dorca Nyamusi
%D 2024
%\ 01/01/2024
%V 20
%N 1
%P 175-200
%! A little more on ideals associated with sublocales
%K frame
%K locale
%K sublocale
%K pointfree function ring
%K Lindel\"{o}f
%K realcompact
%K paracompact
%R 10.48308/cgasa.2023.234093.1456
%X 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{DS1}, 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{DS1} 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 $\mathcal RL$.
%U https://cgasa.sbu.ac.ir/article_104102_2ad4eee6ea89fcd4ab8026764c4c6e92.pdf