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\ʹ{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 $h\colon‎ ‎L\to M$ in $\mathbf{CRegFrm}$‎, ‎$\Zid$ creates a coherent frame‎ ‎homomorphism $\Zid(h)\colon\Zid(\mathcal{R}L)\to\Zid(\mathcal{R}M)$‎ ‎whose right adjoint maps as $(\mathcal{R}h)^{-1}$‎, ‎for the induced‎ ‎ring homomorphism $\mathcal{R}h\colon\mathcal{R}L\to\mathcal{R}M$‎. ‎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. ‎