## Tag `0AS4`

Chapter 10: Commutative Algebra > Section 10.49: Valuation rings

Lemma 10.49.5. Let $I$ be a directed set. Let $(A_i, \varphi_{ij})$ be a system of valuation rings over $I$. Then $A = \mathop{\rm colim}\nolimits A_i$ is a valuation ring.

Proof.It is clear that $A$ is a domain. Let $a, b \in A$. Lemma 10.49.4 tells us we have to show that either $a | b$ or $b | a$ in $A$. Choose $i$ so large that there exist $a_i, b_i \in A_i$ mapping to $a, b$. Then Lemma 10.49.3 applied to $a_i, b_i$ in $A_i$ implies the result for $a, b$ in $A$. $\square$

The code snippet corresponding to this tag is a part of the file `algebra.tex` and is located in lines 11211–11216 (see updates for more information).

```
\begin{lemma}
\label{lemma-colimit-valuation-rings}
Let $I$ be a directed set. Let $(A_i, \varphi_{ij})$
be a system of valuation rings over $I$.
Then $A = \colim A_i$ is a valuation ring.
\end{lemma}
\begin{proof}
It is clear that $A$ is a domain. Let $a, b \in A$.
Lemma \ref{lemma-x-or-x-inverse-valuation-ring} tells us we have
to show that either $a | b$ or $b | a$ in $A$. Choose $i$
so large that there exist $a_i, b_i \in A_i$ mapping to $a, b$.
Then Lemma \ref{lemma-valuation-ring-x-or-x-inverse}
applied to $a_i, b_i$ in $A_i$ implies the result for $a, b$ in $A$.
\end{proof}
```

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