ON MAXIMAL IDEALS OF R∞L | ||
| Journal of Algebraic Systems | ||
| مقاله 4، دوره 6، شماره 1، آذر 2018، صفحه 43-57 اصل مقاله (204.66 K) | ||
| نوع مقاله: Original Manuscript | ||
| شناسه دیجیتال (DOI): 10.22044/jas.2018.6259.1311 | ||
| نویسندگان | ||
| A. A. Estaji* 1؛ A. Mahmoudi Darghadam2 | ||
| 1Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran. Email: aaestaji@hsu.ac.ir and aaestaji@gmail.com | ||
| 2Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran. Email: m.darghadam@yahoo.com | ||
| چکیده | ||
| 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 $\mathcal{R}_{\infty}L$. | ||
| کلیدواژهها | ||
| Frame؛ Compact؛ Maximal ideal؛ Ring of real valued continuous functions | ||
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