In this paper, we introduce a method by which we can find a close connection between the set of primez-ideals of C(X) and the same of C (Y), for some special subset Y of X. For instance, if Y=Coz (f) for some fÎC (X), then there exists a oneto-one correspondence between the set of primez-ideals of C (Y) and the set of primez-ideals of C (X) not containing f. Moreover, considering these relations, we obtain some new characterizations of classical concepts in the context of C (X). For example, X is an F-space if and only if the extension F: bY®bX of the identity map i: Y ® X is one-to-one, for each z-embedded subspace Y of X. Supposing p is a non-isolated Gd-point in X and Y=X \ {p}, we prove that Mp (X) contains no non-trivial maximal z-ideal if and only if pÎ bX is a quasi P-point if and only if each point of bY\ Y is a P-point with respect to Y.