## Tag `032M`

Chapter 10: Commutative Algebra > Section 10.155: Japanese rings

Lemma 10.155.11. A Noetherian domain of characteristic zero is N-1 if and only if it is N-2 (i.e., Japanese).

Proof.This is clear from Lemma 10.155.8 since every field extension in characteristic zero is separable. $\square$

The code snippet corresponding to this tag is a part of the file `algebra.tex` and is located in lines 42797–42801 (see updates for more information).

```
\begin{lemma}
\label{lemma-domain-char-zero-N-1-2}
A Noetherian domain of characteristic zero is N-1 if and only if
it is N-2 (i.e., Japanese).
\end{lemma}
\begin{proof}
This is clear from
Lemma \ref{lemma-Noetherian-normal-domain-finite-separable-extension}
since every field extension in characteristic zero is separable.
\end{proof}
```

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