Maximal Ideals in Functions Rings with a Countable Pointfree Image

Consider the subring RcL of continuous real-valued functions defined on a frame L, comprising functions with a countable pointfree image. We present some useful properties of RcL. We establish that both RcL and its bounded part, Rc^*L, are clean rings for any frame L. We show that, for any completely regular frame L, the zc-ideals of RcL are contractions of the z-ideals of RL. This leads to the conclusion that maximal ideals (or prime zc-ideals) of RcL correspond precisely to the contractions of those of RL. We introduce the Oc- and Mc-ideals of RcL. By using Mc-ideals, we characterize the maximal ideals of RcL, drawing an analogy with the Gelfand-Kolmogoroff theorem for the maximal ideals of Cc (X). We demonstrate that fixed maximal ideals of RcL 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 Rc^*L, leading to a one-to-one correspondence between these ideals and the points of L, the Stone-Cech compactification of L, when L is a strongly zero-dimensional frame. Finally, we establish that ₀L, the Banaschewski compactification of a zero-dimensional L, is isomorphic to the frames of the structure spaces of RcL, Rc (₀L), and R (₀L).