## Tag `032F`

Chapter 10: Commutative Algebra > Section 10.155: Japanese rings

Definition 10.155.1. Let $R$ be a domain with field of fractions $K$.

- We say $R$ is
N-1if the integral closure of $R$ in $K$ is a finite $R$-module.- We say $R$ is
N-2orJapaneseif for any finite extension $K \subset L$ of fields the integral closure of $R$ in $L$ is finite over $R$.

The code snippet corresponding to this tag is a part of the file `algebra.tex` and is located in lines 42562–42575 (see updates for more information).

```
\begin{definition}
\label{definition-N}
\begin{reference}
\cite[Chapter 0, Definition 23.1.1]{EGA}
\end{reference}
Let $R$ be a domain with field of fractions $K$.
\begin{enumerate}
\item We say $R$ is {\it N-1} if the integral closure of $R$ in $K$
is a finite $R$-module.
\item We say $R$ is {\it N-2} or {\it Japanese} if for any finite
extension $K \subset L$ of fields the integral closure of $R$ in $L$
is finite over $R$.
\end{enumerate}
\end{definition}
```

## References

[EGA, Chapter 0, Definition 23.1.1]

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