Exercise 9, § 4

(a) Show that the prime spectrum of a commutative ring is a Kolmogoroff space (General Topology, Chapter I, § 1, Exercise 2), in which every irreducible component admits a (unique) generic point. (b) Let A be an integral domain and ξ = (0) the generic point of X = Spec(A). For the point ξ to be isolated in X, it is necessary and sufficient that the intersection of the prime ideals ≠ 0 of A be an ideal ≠ 0. Give an example of a local ring with this property.