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Tag 032K

Chapter 10: Commutative Algebra > Section 10.155: Japanese rings

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 42578–42583 (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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