Abstract: Let Cc(X) = {f ∈ C(X) : |f(X)| ≤ ℵ0} and CF (X) be the socle of C(X). Similar to C(X) it is shown that uniform ideals and minimal ideals in Cc(X) coincide. Essential ideals in Cc(X) via a topological property are characterized. We define essential zc-filter and it is shown that essential zc-filters behave like zc-ultrafilters and prime zc-filters. It is shown that X is a CP-space if and only if every essential ideal in Cc(X) is a zc-ideal. Cc(X) enjoys most of the important properties of C(X). We observe that if X is an almost discrete space, then CF (X) is an essential ideal in Cc(X). For essentiality of CF (X) in some subrings of C(X) the cardinality of I(X) is important. In particular, if |I(X)| < ∞, then CF (X) is not essential in any subring of C(X).
