Intersections of essential (resp. free) maximal ideals of C(X)C(X)

In this paper, we present a new characterization for the intersection of all essential (resp., free) maximal ideals of C(X)C(X). Thereafter topological spaces X   for which CF(X)CF(X) and CK(X)CK(X) are equal to the intersection of all free (resp. essential) maximal ideals are characterized. Finally we investigate similar situations in an arbitrary commutative semiprimitive ring.