INTERSECTION OF ESSENTIAL IDEALS IN THE RING OF REAL-VALUED CONTINUOUS FUNCTIONS ON A FRAME
Publish place: Journal of Algebraic Systems، Vol: 5، Issue: 2
Publish Year: 1397
نوع سند: مقاله ژورنالی
زبان: English
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شناسه ملی سند علمی:
JR_JAS-5-2_006
تاریخ نمایه سازی: 18 تیر 1398
Abstract:
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 $Sigma L.$
Keywords:
Frame , essential ideal , socle , zero sets in pointfree topology , ring of real-valued continuous functions on a frame
Authors
A. A. Estaji
Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabze- var, Iran.
A. Gh. Karimi Feizabadi
Department of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan,
M. Abedi
Esfarayen University of Technology, Esfarayen, Iran.