Corollary 2

Let $A$ be a ring and $B = A[X]$ the ring of polynomials in one indeterminate with coefficients in $A$. One has :

The first inequality has already been proved (§ 1, No. 3, example 4). Let us prove the second one. For every prime ideal $p$ of $A$, the ring $B \otimes_A \kappa(p)$, isomorphic to $\kappa(p)[X]$, is principal and is not a field, hence is of dimension 1 (§ 1, No. 3, example 2), and the inequality follows from prop. 3.

However, whatever the integers n and q with n + 1 ≤ q ≤ 2n + 1 may be, there exists a ring A of dimension n such that dim(A[X]) = q (see p. 84, exerc. 7).