Shahrood University of TechnologyJournal of Algebraic Systems2345-51285220180101INTERSECTION OF ESSENTIAL IDEALS IN THE RING OF REAL-VALUED CONTINUOUS FUNCTIONS ON A FRAME149161109910.22044/jas.2017.5302.1272ENA. A. EstajiFaculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabze-
var, Iran.A. Gh. Karimi FeizabadiDepartment of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan,M. AbediEsfarayen University of Technology, Esfarayen, Iran.Journal Article20170117A 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 $Sigma L.$http://jas.shahroodut.ac.ir/article_1099_9dfc8c0b4509368b035dd36aa8a9f7c3.pdf