Shahrood University of Technology Journal of Algebraic Systems 2345-5128 6 1 2018 09 01 ON MAXIMAL IDEALS OF R∞L 43 57 1254 10.22044/jas.2018.6259.1311 EN A. A. Estaji Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran. Email: aaestaji@hsu.ac.ir and aaestaji@gmail.com A. Mahmoudi Darghadam Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran. Email: m.darghadam@yahoo.com Journal Article 2017 09 26 Let $L$ be a completely regular frame and $\mathcal{R}L$ be the ring of real-valued continuous functions<br /> on $L$.<br /> We consider the set $$\mathcal{R}_{\infty}L = \{\varphi \in \mathcal{R} L : \uparrow \varphi( \dfrac{-1}{n}, \dfrac{1}{n})<br /> \mbox{ is a compact frame for any $n \in \mathbb{N}$}\}.$$<br /> Suppose that $C_{\infty} (X)$ is the family of all functions $f \in C(X)$ for which the<br /> set $\{x \in X: |f(x)|\geq \dfrac{1}{n} \}$<br /> is compact, for every $n \in \mathbb{N}$.<br /> Kohls has shown that $C_{\infty} (X)$ is precisely the intersection<br /> of all the free maximal ideals of $C^{*}(X)$.<br /> The aim of this paper is to<br /> extend this result to<br /> the real continuous functions on a<br /> frame and hence we show that $\mathcal{R}_{\infty}L$ is precisely the intersection<br /> of all the free maximal ideals of $\mathcal R^{*}L$.<br /> This result is used to characterize the maximal ideals in $\mathcal{R}_{\infty}L$. https://jas.shahroodut.ac.ir/article_1254_45a4703f3fc3297b882c27efeed5d7db.pdf