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.