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In the present paper we give a duality between a special type of ideals of subalgebras of $C\left(X\right)$ containing $C^{*}\left(X\right)$ and $z$-filters of $\beta X$ by generalization of the notion $z$-ideal of $C\left(X\right)$. We also use it to establish some intersecting properties of prime ideals lying between $C^{*}\left(X\right)$ and $C\left(X\right)$. For instance we may mention that such an ideal becomes prime if and only if it contains a prime ideal. Another interesting one is that for such an ideal the residue class ring is totally ordered if and only if it is prime.