Integral Closure is Integrally Closed

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Theorem

Let $A \subseteq B$ be an extension of commutative rings with unity.

Let $C$ be the integral closure of $A$ in $B$.

Then $C$ is integrally closed.


Proof

Suppose $x \in B$ is integral over $C$.

Certainly $C$ is integral over $A$, so by Transitivity of Integrality, $C[x]$ is integral over $A$.

In particular, $x$ is integral over $A$, so $x \in C$.

$\blacksquare$