# The Stacks Project

## Tag 032F

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

1. We say $R$ is N-1 if the integral closure of $R$ in $K$ is a finite $R$-module.
2. We say $R$ is N-2 or Japanese if 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]

Comment #2747 by Ariyan Javanpeykar on August 2, 2017 a 12:11 am UTC

A reference for this definition: EGA IV_0, Definition 23.1.1

Comment #2862 by Johan (site) on October 4, 2017 a 5:42 pm UTC

There are also 3 comments on Section 10.155: Commutative Algebra.

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