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Tag 00NK

Chapter 10: Commutative Algebra > Section 10.104: Catenary rings

Lemma 10.104.7. Any quotient of a catenary ring is catenary. Any quotient of a Noetherian universally catenary ring is universally catenary.

Proof. Let $A$ be a ring and let $I \subset A$ be an ideal. The description of $\mathop{\mathrm{Spec}}(A/I)$ in Lemma 10.16.7 shows that if $A$ is catenary, then so is $A/I$. The second statement is a special case of Lemma 10.104.5. $\square$

    The code snippet corresponding to this tag is a part of the file algebra.tex and is located in lines 24291–24296 (see updates for more information).

    \begin{lemma}
    \label{lemma-quotient-catenary}
    Any quotient of a catenary ring is catenary.
    Any quotient of a Noetherian universally catenary ring is
    universally catenary.
    \end{lemma}
    
    \begin{proof}
    Let $A$ be a ring and let $I \subset A$ be an ideal.
    The description of $\Spec(A/I)$ in Lemma \ref{lemma-spec-closed}
    shows that if $A$ is catenary, then so is $A/I$.
    The second statement is a special case of
    Lemma \ref{lemma-universally-catenary}.
    \end{proof}

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