ON MAXIMAL IDEALS OF R∞L
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 $$mathcal{R}_{infty}L = {varphi in mathcal{R} L : uparrow varphi( dfrac{-1}{n}, dfrac{1}{n})<br /> mbox{ is a compact frame for any $n in...
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| Format: | Article |
| Language: | English |
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Shahrood University of Technology
2018-09-01
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| Series: | Journal of Algebraic Systems |
| Subjects: | |
| Online Access: | http://jas.shahroodut.ac.ir/article_1254_45a4703f3fc3297b882c27efeed5d7db.pdf |
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| author | A. A. Estaji A. Mahmoudi Darghadam |
| author_facet | A. A. Estaji A. Mahmoudi Darghadam |
| author_sort | A. A. Estaji |
| collection | DOAJ |
| description | Let $L$ be a completely regular frame and $mathcal{R}L$ be the ring of real-valued continuous functions<br /> on $L$.<br /> We consider the set $$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 $n in mathbb{N}$}}.$$<br /> Suppose that $C_{infty} (X)$ is the family of all functions $f in C(X)$ for which the<br /> set ${x in X: |f(x)|geq dfrac{1}{n} }$<br /> is compact, for every $n in mathbb{N}$.<br /> Kohls has shown that $C_{infty} (X)$ is precisely the intersection<br /> of all the free maximal ideals of $C^{*}(X)$.<br /> The aim of this paper is to<br /> extend this result to<br /> the real continuous functions on a<br /> frame and hence we show that $mathcal{R}_{infty}L$ is precisely the intersection<br /> of all the free maximal ideals of $mathcal R^{*}L$.<br /> This result is used to characterize the maximal ideals in $mathcal{R}_{infty}L$. |
| first_indexed | 2024-04-13T17:14:21Z |
| format | Article |
| id | doaj.art-05580130e55f4e1989bd38eddae316af |
| institution | Directory Open Access Journal |
| issn | 2345-5128 2345-511X |
| language | English |
| last_indexed | 2024-04-13T17:14:21Z |
| publishDate | 2018-09-01 |
| publisher | Shahrood University of Technology |
| record_format | Article |
| series | Journal of Algebraic Systems |
| spelling | doaj.art-05580130e55f4e1989bd38eddae316af2022-12-22T02:38:11ZengShahrood University of TechnologyJournal of Algebraic Systems2345-51282345-511X2018-09-0161435710.22044/jas.2018.6259.13111254ON MAXIMAL IDEALS OF R∞LA. A. Estaji0A. Mahmoudi Darghadam1Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran. Email: aaestaji@hsu.ac.ir and aaestaji@gmail.comFaculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran. Email: m.darghadam@yahoo.comLet $L$ be a completely regular frame and $mathcal{R}L$ be the ring of real-valued continuous functions<br /> on $L$.<br /> We consider the set $$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_45a4703f3fc3297b882c27efeed5d7db.pdfframecompactmaximal idealring of real valued continuous functions |
| spellingShingle | A. A. Estaji A. Mahmoudi Darghadam ON MAXIMAL IDEALS OF R∞L Journal of Algebraic Systems frame compact maximal ideal ring of real valued continuous functions |
| title | ON MAXIMAL IDEALS OF R∞L |
| title_full | ON MAXIMAL IDEALS OF R∞L |
| title_fullStr | ON MAXIMAL IDEALS OF R∞L |
| title_full_unstemmed | ON MAXIMAL IDEALS OF R∞L |
| title_short | ON MAXIMAL IDEALS OF R∞L |
| title_sort | on maximal ideals of r∞l |
| topic | frame compact maximal ideal ring of real valued continuous functions |
| url | http://jas.shahroodut.ac.ir/article_1254_45a4703f3fc3297b882c27efeed5d7db.pdf |
| work_keys_str_mv | AT aaestaji onmaximalidealsofrl AT amahmoudidarghadam onmaximalidealsofrl |