ON MAXIMAL IDEALS OF R∞L

Abstract

Let $L$ be a completely regular frame and $mathcal{R}L$ be the ring of real-valued continuous functions on $L$. We consider the set $$mathcal{R}_{infty}L = {varphi in mathcal{R} L : uparrow varphi( dfrac{-1}{n}, dfrac{1}{n}) mbox{ is a compact frame for any $n in mathbb{N}$}}.$$ Suppose that $C_{infty} (X)$ is the family of all functions $f in C(X)$ for which the set ${x in X: |f(x)|geq dfrac{1}{n} }$ is compact, for every $n in mathbb{N}$. Kohls has shown that $C_{infty} (X)$ is precisely the intersection of all the free maximal ideals of $C^{*}(X)$. The aim of this paper is to extend this result to the real continuous functions on a frame and hence we show that $mathcal{R}_{infty}L$ is precisely the intersection of all the free maximal ideals of $mathcal R^{*}L$. This result is used to characterize the maximal ideals in $mathcal{R}_{infty}L$.