## Tag `0308`

Chapter 10: Commutative Algebra > Section 10.35: Finite and integral ring extensions

Lemma 10.35.16. Let $A \to B \to C$ be ring maps. Let $B'$ be the integral closure of $A$ in $B$, let $C'$ be the integral closure of $B'$ in $C$. Then $C'$ is the integral closure of $A$ in $C$.

Proof.Omitted. $\square$

The code snippet corresponding to this tag is a part of the file `algebra.tex` and is located in lines 7370–7376 (see updates for more information).

```
\begin{lemma}
\label{lemma-integral-closure-transitive}
Let $A \to B \to C$ be ring maps.
Let $B'$ be the integral closure of $A$ in $B$,
let $C'$ be the integral closure of $B'$ in $C$. Then
$C'$ is the integral closure of $A$ in $C$.
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
```

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