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