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.