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

Abstract In this paper, we present a new characterization for the intersection of all essential (resp., free) maximal ideals of C ( X ) . Thereafter topological spaces X for which C F ( X ) and C K ( 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.