Intersection of essential ideals in $C(X)$

The infinite intersection of essential ideals in any ring may not be an essential ideal, this intersection may even be zero. By the topological characterization of the socle by Karamzadeh and Rostami (Proc. Amer. Math.Soc. 93 (1985), 179-184), and the topological characterization of essential ideals in Proposition 2.1, it is easy to see that every intersection of essential ideals of $C(X)$ is an essential ideal if and only if the set of isolated points of $X$ is dense in $X$. Motivated by this result in $C(X)$, we study the essentiallity of the intersection of essential ideals for topological spaces which may have no isolated points. In particular, some important ideals $C_K(X)$ and $C_\infty (X)$, which are the intersection of essential ideals, are studied further and their essentiallity is characterized. Finally a question raised by Karamzadeh and Rostami, namely when the socle of $C(X)$ and the ideal of $C_K(X)$ coincide, is answered.