## Tag `032G`

Chapter 10: Commutative Algebra > Section 10.155: Japanese rings

Lemma 10.155.3. Let $R$ be a domain. If $R$ is N-1 then so is any localization of $R$. Same for N-2.

Proof.These statements hold because taking integral closure commutes with localization, see Lemma 10.35.11. $\square$

The code snippet corresponding to this tag is a part of the file `algebra.tex` and is located in lines 42600–42605 (see updates for more information).

```
\begin{lemma}
\label{lemma-localize-N}
Let $R$ be a domain.
If $R$ is N-1 then so is any localization of $R$.
Same for N-2.
\end{lemma}
\begin{proof}
These statements hold because taking integral closure commutes
with localization, see Lemma \ref{lemma-integral-closure-localize}.
\end{proof}
```

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