TY - JOUR ID - 1099 TI - INTERSECTION OF ESSENTIAL IDEALS IN THE RING OF REAL-VALUED CONTINUOUS FUNCTIONS ON A FRAME JO - Journal of Algebraic Systems JA - JAS LA - en SN - 2345-5128 AU - Estaji, A. A. AU - Karimi Feizabadi, A. Gh. AU - Abedi, M. AD - Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabze- var, Iran. AD - Department of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan, AD - Esfarayen University of Technology, Esfarayen, Iran. Y1 - 2018 PY - 2018 VL - 5 IS - 2 SP - 149 EP - 161 KW - Frame KW - essential ideal KW - socle KW - zero sets in pointfree topology KW - ring of real-valued continuous functions on a frame DO - 10.22044/jas.2017.5302.1272 N2 - 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 $\Sigma L.$ UR - https://jas.shahroodut.ac.ir/article_1099.html L1 - https://jas.shahroodut.ac.ir/article_1099_9dfc8c0b4509368b035dd36aa8a9f7c3.pdf ER -