Estaji, A., Karimi Feizabadi, A., Abedi, M. (2018). INTERSECTION OF ESSENTIAL IDEALS IN THE RING OF REAL-VALUED CONTINUOUS FUNCTIONS ON A FRAME. Journal of Algebraic Systems, 5(2), 149-161. doi: 10.22044/jas.2017.5302.1272
A. A. Estaji; A. Gh. Karimi Feizabadi; M. Abedi. "INTERSECTION OF ESSENTIAL IDEALS IN THE RING OF REAL-VALUED CONTINUOUS FUNCTIONS ON A FRAME". Journal of Algebraic Systems, 5, 2, 2018, 149-161. doi: 10.22044/jas.2017.5302.1272
Estaji, A., Karimi Feizabadi, A., Abedi, M. (2018). 'INTERSECTION OF ESSENTIAL IDEALS IN THE RING OF REAL-VALUED CONTINUOUS FUNCTIONS ON A FRAME', Journal of Algebraic Systems, 5(2), pp. 149-161. doi: 10.22044/jas.2017.5302.1272
Estaji, A., Karimi Feizabadi, A., Abedi, M. INTERSECTION OF ESSENTIAL IDEALS IN THE RING OF REAL-VALUED CONTINUOUS FUNCTIONS ON A FRAME. Journal of Algebraic Systems, 2018; 5(2): 149-161. doi: 10.22044/jas.2017.5302.1272
INTERSECTION OF ESSENTIAL IDEALS IN THE RING OF REAL-VALUED CONTINUOUS FUNCTIONS ON A FRAME
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.
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 $\Sigma L.$
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