Exercise 11, § 2

Let $A$ be an integral ring and $K$ its field of fractions. Let $(0) \subset p_1 \subset ... \subset p_h$ be a chain of prime ideals of $A$. Now prove that there exists a valuation ring $V$ of $K$ and a chain of prime ideals $(0) \subset q_1 \subset ... \subset q_h$ of $V$ such that $q_i \cap A = p_i$. (We shall proceed by induction on $h$, reducing to the case where $A$ is a local ring and using, for the case $h = 1$, Exercise 7 of VI, § 1.)