Exercise 14, § 4

Let $A$ be a noetherian integral ring, $K$ its field of fractions, $\overline{A}$ its integral closure in a finite extension $E$ of $K$. Then $\overline{A}$ is a Krull ring. (Use Exercise 11, c) and, for proving that every element $f$ of $\overline{A}$, nonzero, belongs only to a finite number of prime ideals of height 1, reduce to the case where $f \in A$, $E = K$, and use Exercises 13, b) and 11, a).)