Exercise 13, § 2

Let $A$ be an integral domain and $K$ its field of fractions. For every integer $k \geq 0$, the following properties are equivalent: (i) Every subring $B$ of $K$ containing $A$ is of dimension $\leq k$. (ii) Every valuation ring $V$ of $K$ containing $A$ is of dimension $\leq k$. (iii) For every choice of $k$ elements $t_1, ..., t_k$ of $K$, one has $\dim(A[t_1, ..., t_k]) \leq k$. (iv) One has $\dim(A[X_1, ..., X_m]) \leq k + m$ for every integer $m \geq 0$.