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 $R_\alpha:=\{r\in\mathbb R \mid {\mathrm{coz}}(\alpha-r)\neq\top\}$ for every $\alpha\in\mathcal R (L).$ By using idempotent elements, it is going to prove that ${{\mathrm{Coz}}}_c[L]:= \{{\mathrm{coz}}(\alpha) \mid \alpha\in\mathcal{C}_c (L) \}$ 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.