Shahrood University of TechnologyJournal of Algebraic Systems2345-51286120180901<p>ON MAXIMAL IDEALS OF R<sub>∞</sub>L</p>4357125410.22044/jas.2018.6259.1311ENA. A. EstajiFaculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.
Email: aaestaji@hsu.ac.ir and aaestaji@gmail.comA. Mahmoudi DarghadamFaculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.
Email: m.darghadam@yahoo.comJournal Article20170926Let $L$ 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 $mathcal{R}_{infty}L$.http://jas.shahroodut.ac.ir/article_1254_b7ff14694dc7c2f170a0a9a6677d196b.pdf