Maximal Ideals in Functions Rings with a Countable Pointfree Image

TL;DR

This paper investigates the structure of maximal ideals in rings of functions with countable pointfree images on frames, establishing their properties, correspondences with points in various compactifications, and analogies with classical theorems.

Contribution

It introduces and characterizes maximal ideals in subrings of continuous functions on frames with countable pointfree images, extending classical results to a frame-theoretic context.

Findings

Maximal ideals correspond to points in zero-dimensional frames.

Fixed maximal ideals are in one-to-one correspondence with points of the frame.

The structure space of the ring is isomorphic to the Banaschewski compactification.

Abstract

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 $\mathcal{R}(\beta_0L)$.