# The Stacks Project

## Tag 032K

Lemma 10.155.7. Let $R$ be a Noetherian domain, and let $R \subset S$ be a finite extension of domains. If $S$ is N-1, then so is $R$. If $S$ is N-2, then so is $R$.

Proof. Omitted. (Hint: Integral closures of $R$ in extension fields are contained in integral closures of $S$ in extension fields.) $\square$

The code snippet corresponding to this tag is a part of the file algebra.tex and is located in lines 42622–42627 (see updates for more information).

\begin{lemma}
\label{lemma-finite-extension-N-2}
Let $R$ be a Noetherian domain, and let $R \subset S$ be a
finite extension of domains. If $S$ is N-1, then so is $R$.
If $S$ is N-2, then so is $R$.
\end{lemma}

\begin{proof}
Omitted. (Hint: Integral closures of $R$ in extension fields
are contained in integral closures of $S$ in extension fields.)
\end{proof}

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