Document Type : Original Manuscript

Authors

1 Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabze- var, Iran.

2 Department of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan,

3 Esfarayen 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 $\Sigma L.$

Keywords