Proposition 8

Let A be a ring.

a) One has $\dim(A) = \sup_m \dim(A_m) = \sup_m \operatorname{ht}(m)$, where m runs through the set of maximal (resp. prime) ideals of A.

b) Let b be an ideal of A distinct from A, and a an ideal of A contained in b. Then one has $\operatorname{codim}(V(b), V(a)) + \dim(A/b) \leq \dim(A/a)$. In particular, for every ideal b of A distinct from A, one has the inequality $\operatorname{ht}(b) + \dim(A/b) \leq \dim(A)$.

The first assertion follows from Remark 2 of No. 2 and from Proposition 7, b). The second follows from Proposition 3, a) of No. 2 and from the relations $\dim(A/b) = \dim(V(b))$, $\dim(A/a) = \dim(V(a))$.