## Tag `00NK`

Chapter 10: Commutative Algebra > Section 10.104: Catenary rings

Lemma 10.104.6. Any quotient of a (universally) catenary ring is (universally) catenary.

Proof.Let $A$ be a ring and let $I \subset A$ be an ideal. The description of $\mathop{\rm Spec}(A/I)$ in Lemma 10.16.7 shows that if $A$ is catenary, then so is $A/I$. If $A/I \to B$ is of finite type, then $A \to B$ is of finite type. Hence if $A$ is universally catenary, then $B$ is catenary. Combined with Lemma 10.30.1 this proves the lemma. $\square$

The code snippet corresponding to this tag is a part of the file `algebra.tex` and is located in lines 24208–24211 (see updates for more information).

```
\begin{lemma}
\label{lemma-quotient-catenary}
Any quotient of a (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 $A/I$. If $A/I \to B$
is of finite type, then $A \to B$ is of finite type. Hence if $A$
is universally catenary, then $B$ is catenary. Combined with
Lemma \ref{lemma-Noetherian-permanence} this proves the lemma.
\end{proof}
```

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