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Let $C_F\left(X\right)$ be the socle of C(X). It is shown that each prime ideal in $C\left(X\right)/C_F\left(X\right)$ is essential. For each h ∈ C(X), we prove that every prime ideal (resp. z-ideal) of C(X)/(h) is essential if and only if the set Z(h) of zeros of h contains no isolated points (resp. int Z(h) = ∅). It is proved that $dim(C\left(X\right)/C_F\left(X\right))\ge dimC\left(X\right)$, where dim C(X) denotes the Goldie dimension of C(X), and the inequality may be strict. We also give an algebraic characterization of compact spaces with at most a countable number of nonisolated points. For each essential...