In this article, we have characterized ideals in $C(X)$ in which every ideal is also an ideal (a $z$-ideal) of $C(X)$. Motivated by this characterization, we observe that $C_infty(X)$ is a regular ring if and only if every open locally compact $sigma$-compact subset of $X$ is finite. Concerning prime ideals, it is shown that the sum of every two prime (semiprime) ideals of each ideal in $C(X)$ is prime (semiprime) if and only if $X$ is an $F$-space. Concerning maximal ideals of an ideal, we generalize the notion of separability to ideals and we have proved the coincidence of separability of an ideal with dense separability of a subspace of $\beta X$. Finally, we have shown that the Goldie dimension of an ideal $I$ in $C(X)$ coincide with the cellularity of $XsetminusDelta (I)$.