The Cozero part of the pointfree version of $C_c(X)$

Let ${\mathcal{C}}_c(L):=\{\alpha \in \mathcal{R}(L)\mid R_{\alpha }\text{ is a countable subset of }\mathbb{R}\}$, where $R_{\alpha }:=\{r\in \mathbb{R}\mid \mathrm{c}\mathrm{o}\mathrm{z}(\alpha -r)\mathrm{\ne }\mathrm{\top }\}$ for every $\alpha \in \mathcal{R}(L)\mathrm{.}$ By using idempotent elements, it is going to prove that ${\mathrm{C}\mathrm{o}\mathrm{z}}_c[L]:=\{\mathrm{c}\mathrm{o}\mathrm{z}(\alpha )\mid \alpha \in {\mathcal{C}}_c(L)\}$ is a $\sigma$-frame for every completely regular frame $L,$ and from this, we conclude that it is regular, paracompact, perfectly normal and an Alexandroff algebra frame such that each cover of it is shrinkable. Also, we show that $L$ is a zero-dimensional frame if and only if $L$ is a $c$-completely regular frame.