Lemma 10.105.8 (0AUP)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.105: Catenary rings
Lemma 10.105.8 (cite)

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Lemma 10.105.8. Let $R$ be a Noetherian ring.

$R$ is catenary if and only if $R/\mathfrak p$ is catenary for every minimal prime $\mathfrak p$ .
$R$ is universally catenary if and only if $R/\mathfrak p$ is universally catenary for every minimal prime $\mathfrak p$ .

Proof. If $\mathfrak a \subset \mathfrak b$ is an inclusion of primes of $R$ , then we can find a minimal prime $\mathfrak p \subset \mathfrak a$ and the first assertion is clear. We omit the proof of the second. $\square$

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