Document Type : Original Manuscript

Authors

1 Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran. Email: aaestaji@hsu.ac.ir and aaestaji@gmail.com

2 Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran. Email: m.darghadam@yahoo.com

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 $\mathcal{R}_{\infty}L$.

Keywords