ON IDEALS OF IDEALS IN C(X)

In this article, we have characterized ideals in C(X) in which every ideal is also an ideal (a z-ideal) of C(X). Motivated by this characterization, we observe that C1(X) is a regular ring if and only if every open locally compact -compact subset of X is nite. Concerning prime ideals, it is shown that the sum of ev- ery two prime (semiprime) ideals of each ideal in C(X) is prime (semiprime) if and only if X is an F-space. Concerning maximal ideals of an ideal, we generalize the notion of separability to ideals and we have proved the coincidence of separability of an ideal with dense separability of a subspace of X. Finally, we have shown that the Goldie dimension of an ideal I in C(X) coincide with the cellularity of X n Δ(I).