INTERSECTION OF ESSENTIAL IDEALS IN THE RING OF REAL-VALUED CONTINUOUS FUNCTIONS ON A FRAME | ||
| Journal of Algebraic Systems | ||
| مقاله 6، دوره 5، شماره 2، فروردین 2018، صفحه 149-161 اصل مقاله (211.08 K) | ||
| نوع مقاله: Original Manuscript | ||
| شناسه دیجیتال (DOI): 10.22044/jas.2017.5302.1272 | ||
| نویسندگان | ||
| A. A. Estaji1؛ A. Gh. Karimi Feizabadi2؛ M. Abedi* 3 | ||
| 1Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabze- var, Iran. | ||
| 2Department of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan, | ||
| 3Esfarayen University of Technology, Esfarayen, Iran. | ||
| چکیده | ||
| A frame $L$ is called {\it coz-dense} if $\Sigma_{coz(\alpha)}=\emptyset$ implies $\alpha=\mathbf 0$. Let $\mathcal RL$ be the ring of real-valued continuous functions on a coz-dense and completely regular frame $L$. We present a description of the socle of the ring $\mathcal RL$ based on minimal ideals of $\mathcal RL$ and zero sets in pointfree topology. We show that socle of $\mathcal RL$ is an essential ideal in $\mathcal RL$ if and only if the set of isolated points of $ \Sigma L$ is dense in $ \Sigma L$ if and only if the intersection of any family of essential ideals is essential in $\mathcal RL$. Besides, the counterpart of some results in the ring $C(X)$ is studied for the ring $\mathcal RL$. For example, an ideal $E$ of $\mathcal RL$ is an essential ideal if and only if $\bigcap Z[E]$ is a nowhere dense subset of $\Sigma L.$ | ||
| کلیدواژهها | ||
| Frame؛ essential ideal؛ socle؛ zero sets in pointfree topology؛ ring of real-valued continuous functions on a frame | ||
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آمار تعداد مشاهده مقاله: 698 تعداد دریافت فایل اصل مقاله: 1,279 |
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