Exercise 10, § 1

It is said that a ring A is equidimensional if all the irreducible components of Spec(A) have the same dimension. Let A be a ring and let n be an integer. The following conditions are equivalent: a) every chain of prime ideals of A is contained in a chain of length n; b) for every maximal ideal m of A, the ring A_m is equidimensional, catenary and of dimension n.