Let $A(X)$ be a subring of $C(X)$ that contains $C^*(X)$. In Redlin and Watson (1987) and in Panman et al. (2012), correspondences ${\cal Z}_A$ and $\frak Z_A$ are defined between ideals in $A(X)$ and $z$-filters on $X$, and it is shown that these extend the well-known correspondences studied separately for $C^*(X)$ and $C(X)$, respectively, to any intermediate ring $A(X)$. Moreover, the inverse map $\cal Z^{-1}_A$ sets up a one-one correspondence between the maximal ideals of $A(X)$ and the $z$-ultrafilters on $X$. In this paper, first, we characterize essential ideals in $A(X)$. Afterwards, we show that $\cal Z^{-1}_A$ maps essential (resp., free) $z$-filters on $X$ to essential (resp., free) ideals in $A(X)$ and $\frak Z^{-1}_A$ maps essential $\frak Z_A$-filters to essential ideals. Similar to $C(X)$ we observe that the intersection of all essential minimal prime ideals in $A(X)$ is equal to the socle of $A(X)$. Finally, we give a new characterization for the intersection of all essential maximal ideals of $A(X)$.