Abstract Let Cc(X) (resp. CF (X)) denote the subring of C(X) consisting of functions with countable (resp. finite) image and CF (X) be the socle of C(X). We characterize spaces X with C∗(X) = Cc(X), which generalizes a celebrated result due to Rudin, Pelczynnski, and Semadeni. Two ( zero-dimensional compact spaces X, Y are homeomorphic if and only if Cc(X)∼=Cc(Y ) .(resp. CF (X)∼=CF (Y ) The spaces X for which Cc(X) = CF (X) are characterized. The socles of Cc(X) (resp. C CF (X), which are observed to be the same, are topologically characterized and spaces X for which this socle coincides with C^*F (X) are determined, too. A certain well-known lgebraic property of C(X), where X is real compact, is extended to Cc(X). In contrast to the fact that CF (X) is never prime in C(X), we characterize spaces X for which CF (X) is a prime ideal in Cc(X). It is observed for these spaces, Cc(X) coincides with its own socle (a fact, which is never true for C(X), unless X is finite). Finally, we show that a space X is the one-point compactification of a discrete space if and only if C^*F (X) is a unique proper essential ideal in .(C^*F (X
