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Tag 032M

Chapter 10: Commutative Algebra > Section 10.155: Japanese rings

Lemma 10.155.11. A Noetherian domain of characteristic zero is N-1 if and only if it is N-2 (i.e., Japanese).

Proof. This is clear from Lemma 10.155.8 since every field extension in characteristic zero is separable. $\square$

    The code snippet corresponding to this tag is a part of the file algebra.tex and is located in lines 42691–42695 (see updates for more information).

    \begin{lemma}
    \label{lemma-domain-char-zero-N-1-2}
    A Noetherian domain of characteristic zero is N-1 if and only if
    it is N-2 (i.e., Japanese).
    \end{lemma}
    
    \begin{proof}
    This is clear from
    Lemma \ref{lemma-Noetherian-normal-domain-finite-separable-extension}
    since every field extension in characteristic zero is separable.
    \end{proof}

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