## Tag `0335`

Chapter 10: Commutative Algebra > Section 10.156: Nagata rings

Proposition 10.156.16. The following types of rings are Nagata and in particular universally Japanese:

- fields,
- Noetherian complete local rings,
- $\mathbf{Z}$,
- Dedekind domains with fraction field of characteristic zero,
- finite type ring extensions of any of the above.

Proof.The Noetherian complete local ring case is Lemma 10.156.8. In the other cases you just check if $R/\mathfrak p$ is N-2 for every prime ideal $\mathfrak p$ of the ring. This is clear whenever $R/\mathfrak p$ is a field, i.e., $\mathfrak p$ is maximal. Hence for the Dedekind ring case we only need to check it when $\mathfrak p = (0)$. But since we assume the fraction field has characteristic zero Lemma 10.155.11 kicks in. $\square$

The code snippet corresponding to this tag is a part of the file `algebra.tex` and is located in lines 43538–43548 (see updates for more information).

```
\begin{proposition}
\label{proposition-ubiquity-nagata}
The following types of rings are Nagata and in particular universally Japanese:
\begin{enumerate}
\item fields,
\item Noetherian complete local rings,
\item $\mathbf{Z}$,
\item Dedekind domains with fraction field of characteristic zero,
\item finite type ring extensions of any of the above.
\end{enumerate}
\end{proposition}
\begin{proof}
The Noetherian complete local ring case is
Lemma \ref{lemma-Noetherian-complete-local-Nagata}.
In the other cases you just check if $R/\mathfrak p$ is N-2 for every
prime ideal $\mathfrak p$ of the ring. This is clear whenever
$R/\mathfrak p$ is a field, i.e., $\mathfrak p$ is maximal.
Hence for the Dedekind ring case we only need to check it when
$\mathfrak p = (0)$. But since we assume the fraction field has
characteristic zero Lemma \ref{lemma-domain-char-zero-N-1-2} kicks in.
\end{proof}
```

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