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). IF X IS ANY TOPOLOGICAL SPACE THERE IS A ZERO-DIMENSIONAL SPACEY SUCH THAT CC(X) @ CC(Y). 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 WELL-KNOWN ALGEBRAIC PROPERTY OF C(X), WHERE X IS REALCOMPACT, 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)). FINALLY, WE SHOW THAT A SPACE X IS THE ONE-POINT COMPACTIFICATION OF A DISCRETE SPACE IF AND ONLY IF CF(X) IS A UNIQUE PROPER ESSENTIAL IDEAL IN CF(X), SEE [9], [10]. A SIMILAR CHARACTERIZATION, AS THE GELFAND-KOLMOGOROFF THEOREM FOR THE MAXIMAL IDEALS IN C(X), IS GIVEN FOR THE MAXIMAL IDEALS OF CC(X), SEE [4]. THE SUBALGEBRA LC(X) = {¦ Í C(X): C¦ = X} OF C(X), WHERE C¦ IS THE UNION OF ALL OPEN SUBSETS U Í  X SUCH THAT |¦ (U)| £ À0, WHICH IS CC(X) Í LC(X) Í C(X), SEE [13].