Let C_c(X) = {f ЄC(X) : f(X) is countable}. Similar to C(X) it is observed that the sum of any collection of semiprime (resp. prime) ideals in the ring C_c(X) is either C_c(X) or a semiprime (resp. prime) ideal in C_c(X). For an ideal I in C_c(X), it is observed that I and Ī have the same largest z_c-ideal. If X is any topological space, we show that there is a zero-dimensional space Y such that C_c(X) ≈C_c(Y ). Consequently, if X has only countable number of components, then C_c(X) ≈C(Y ) for some zero-dimensional space Y . Spaces X for which C_c(X) is regular (called CP-spaces) are characterized both algebraically and topologically and it is shown that P-spaces and CP-spaces coincide when X is zero-dimensional. In contrast to C*(X), we observe that C_c(X) enjoys the algebraic properties of regularity, N_0 - selfinjectivity and some others, whenever C(X) has these properties. Finally an example of a space X such that C_c(X) is not isomorphic to any C(Y ) is given.
