## Tag `052L`

Chapter 10: Commutative Algebra > Section 10.49: Valuation rings

Lemma 10.49.6. Let $K \subset L$ be an extension of fields. If $B \subset L$ is a valuation ring, then $A = K \cap B$ is a valuation ring.

Proof.We can replace $L$ by $f.f.(B)$ and $K$ by $K \cap f.f.(B)$. Then the lemma follows from a combination of Lemmas 10.49.3 and 10.49.4. $\square$

The code snippet corresponding to this tag is a part of the file `algebra.tex` and is located in lines 11228–11232 (see updates for more information).

```
\begin{lemma}
\label{lemma-valuation-ring-cap-field}
Let $K \subset L$ be an extension of fields. If $B \subset L$
is a valuation ring, then $A = K \cap B$ is a valuation ring.
\end{lemma}
\begin{proof}
We can replace $L$ by $f.f.(B)$ and $K$ by $K \cap f.f.(B)$.
Then the lemma follows from a combination of
Lemmas \ref{lemma-valuation-ring-x-or-x-inverse} and
\ref{lemma-x-or-x-inverse-valuation-ring}.
\end{proof}
```

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