Discrete \(z\)-filters and rings of analytic functions

Consider rings of single variable real analytic or complex entire functions, denoted by \(\mathbb{K}\langle z\rangle\). We study "discrete \(z\)-filters" on \(\mathbb{K}\) and their connections with the space of maximal ideals of \(\mathbb{K}\langle z\rangle\), which we characterize as a compact \(T_1\) space \(\theta \mathbb{K}\) of discrete \(z\)-ultrafilters on \(\mathbb{K}\). We show that \(\theta \mathbb{K}\) is a bijective continuous image of \(\beta \mathbb{K} \setminus Q(\mathbb{K})\), where \(Q(\mathbb{K})\) is the set of far points of \(\beta \mathbb{K}\). \(\theta \mathbb{K}\) turns out to be the Wallman compactification of the canonically embedded image of \(\mathbb{K}\) inside \(\theta\mathbb{K}\). Using our characterization of \(\theta\mathbb{K}\), we derive a Gelfand-Kolmogorov characterization of maximal ideals of \(\mathbb{K}\langle z\rangle\) and show that the Krull dimension of \(\mathbb{K}\langle z\rangle\) is at least \(c\). We also establish the existence of a chain of prime \(z\)-filters on \(\mathbb{K}\) consisting of at least \(2^c\) many elements.