on $z$-ideals of pointfree function rings

let $l$ be a completely regular frame and $mathcal{r}l$ be the ‎ring of continuous real-valued functions on $l$‎. ‎we show that the‎ ‎lattice $zid(mathcal{r}l)$ of $z$-ideals of $mathcal{r}l$ is a‎ ‎normal coherent yosida frame‎, ‎which extends the corresponding $c(x)$‎ ‎result of mart'{i}nez and zenk‎. ‎this we do by exhibiting‎ ‎$zid(mathcal{r}l)$ as a quotient of $rad(mathcal{r}l)$‎, ‎the‎ ‎frame of radical ideals of $mathcal{r}l$‎. ‎the saturation quotient‎ ‎of $zid(mathcal{r}l)$ is shown to be isomorphic to the‎ ‎stone-v{c}ech compactification of $l$‎. ‎given a morphism $hcolon ‎lto m$ in $mathbf{cregfrm}$‎, ‎$zid$ creates a coherent frame‎ ‎homomorphism $zid(h)colonzid(mathcal{r}l)tozid(mathcal{r}m)$‎ ‎whose right adjoint maps as $(mathcal{r}h)^{-1}$‎, ‎for the induced‎ ‎ring homomorphism $mathcal{r}hcolonmathcal{r}ltomathcal{r}m$‎.‎thus‎, ‎$zid(h)$ is an $s$-map‎, ‎in the sense of mart`{i}nez cite{mar1}‎, ‎precisely when‎ ‎$mathcal{r}(h)$ contracts maximal ideals to maximal ideals‎.