  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.
