Maximal Ideals in Functions Rings with a Countable Pointfree Image

Consider the subring $\mathcal{R}_cL$ of continuous real-valued functions defined on a frame $L$, comprising functions with a countable pointfree image. We present some useful properties of $\mathcal{R}_cL$. We establish that both $\mathcal{R}_cL$ and its bounded part, $\mathcal{R}_c^*L$, are clean rings for any frame $L$. We show that, for any completely regular frame $L$, the $z_c$-ideals of $\mathcal{R}_cL$ are contractions of the $z$-ideals of $\mathcal{R}L$. This leads to the conclusion that maximal ideals (or prime $z_c$-ideals) of $\mathcal{R}_cL$ correspond precisely to the contractions of those of $\mathcal{R}L$. We introduce the ${\bf O}_c$- and ${\bf M}_c$-ideals of $\mathcal{R}_cL$. By using ${\bf M}_c$-ideals, we characterize the maximal ideals of $\mathcal{R}_cL$, drawing an analogy with the Gelfand-Kolmogoroff theorem for the maximal ideals of $C_c(X)$. We demonstrate that fixed maximal ideals of $\mathcal{R}_cL$ have a one-to-one correspondence with the points of $L$ in the case where $L$ is a zero-dimensional frame. We describe the maximal ideals of $\mathcal{R}_c^*L$, leading to a one-to-one correspondence between these ideals and the points of $\beta L$, the Stone-\v{C}ech compactification of $L$, when $L$ is a strongly zero-dimensional frame. Finally, we establish that $\beta_0L$, the Banaschewski compactification of a zero-dimensional $L$, is isomorphic to the frames of the structure spaces of $\mathcal{R}_cL$, $\mathcal{R}_c(\beta_0L)$, and $\mathcal{R}(\beta_0L)$.