Correspondences between ideals and $z$-filters for rings of continuous functions between $C^{\ast }$ and $C$

Let $X$ be a completely regular topological space. Let $A(X)$ be a ring of continuous functions between $C^{\ast }(X)$ and $C(X)$, that is, $C^{\ast }(X)\subset A(X)\subset C(X)$. In [9], a correspondence ${\mathcal{Z}}_A$ between ideals of $A(X)$ and $z$-filters on $X$ is defined. Here we show that ${\mathcal{Z}}_A$ extends the well-known correspondence for $C^{\ast }(X)$ to all rings $A(X)$. We define a new correspondence ${\mathcal{Z}}_A$ and show that it extends the well-known correspondence for $C(X)$ to all rings $A(X)$. We give a formula that relates the two correspondences. We use properties of ${\mathcal{Z}}_A$ and ${\mathcal{Z}}_A$ to characterize $C^{\ast }(X)$ and $C(X)$ among all rings $A(X)$. We show that ${\mathcal{Z}}_A$ defines a one-one correspondence between maximal ideals in $A(X)$ and the $z$-ultrafilters in $X$.