@article { author = {Estaji, A. A. and Mahmoudi Darghadam, A.}, title = {<p>ON MAXIMAL IDEALS OF R<sub>∞</sub>L</p>}, journal = {Journal of Algebraic Systems}, volume = {6}, number = {1}, pages = {43-57}, year = {2018}, publisher = {Shahrood University of Technology}, issn = {2345-5128}, eissn = {2345-511X}, doi = {10.22044/jas.2018.6259.1311}, 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 = {Frame,Compact,Maximal ideal,Ring of real valued continuous functions}, url = {https://jas.shahroodut.ac.ir/article_1254.html}, eprint = {https://jas.shahroodut.ac.ir/article_1254_45a4703f3fc3297b882c27efeed5d7db.pdf} }