Corollary 5

Let $A$ be an integral domain, $A'$ its integral closure and $f$ the annihilator of the $A$-module $A'/A$. Suppose that $A'$ is afinitely generated $A$-module. The prime ideals $p \in A$ such that $A_p$ is not integrally closed are those which contain $f$.

This follows immediately from Corollary 4 (ii) applied to $S = A - p$.

Note that under the hypotheses of Corollary 5 $f \neq 0$ since $A'$ is a finitely generated $A$-module and every element of $K/A$ ($K$ the field of fractions of $A$) has an annihilator $\neq 0$.

In algebraic geometry, Corollary 5 and the above remark show that the points where an affine variety $V$ is not normal form a closed set distinct from $V$. \*