Improve definition catenary ring

Thanks to Bjorn Poonen
https://stacks.math.columbia.edu/tag/00NI#comment-5869

algebra.tex

@@ -14099,13 +14099,31 @@ \section{Noetherian local rings}
 \section{Dimension}
 \label{section-dimension}
 
+\noindent
+Please compare with
+Topology, Section \ref{topology-section-krull-dimension}.
+
+\begin{definition}
+\label{definition-chain-primes}
+Let $R$ be a ring. A {\it chain of prime ideals} is a sequence
+$\mathfrak p_0 \subset \mathfrak p_1 \subset \ldots \subset \mathfrak p_n$
+of prime ideals of $R$ such that $\mathfrak p_i \not = \mathfrak p_{i + 1}$
+for $i = 0, \ldots, n - 1$. The {\it length} of this chain of prime
+ideals is $n$.
+\end{definition}
+
+\noindent
+Recall that we have an inclusion reversing bijection between prime
+ideals of a ring $R$ and irreducible closed subsets of $\Spec(R)$,
+see Lemma \ref{lemma-irreducible}.
+
 \begin{definition}
 \label{definition-Krull}
 The {\it Krull dimension} of the ring $R$ is the
 Krull dimension of the topological space $\Spec(R)$, see
 Topology, Definition \ref{topology-definition-Krull}.
 In other words it is the supremum of the integers $n\geq 0$
-such that there exists a chain of prime ideals of length $n$:
+such that $R$ has a chain of prime ideals
 $$
 \mathfrak p_0
 \subset
@@ -14116,6 +14134,7 @@ \section{Dimension}
 \mathfrak p_n, \quad
 \mathfrak p_i \not = \mathfrak p_{i + 1}.
 $$
+of length $n$.
 \end{definition}
 
 \begin{definition}
@@ -24936,12 +24955,16 @@ \section{Cohen-Macaulay rings}
 \section{Catenary rings}
 \label{section-catenary}
 
+\noindent
+Compare with Topology, Section \ref{topology-section-catenary-spaces}.
+
 \begin{definition}
 \label{definition-catenary}
 A ring $R$ is said to be {\it catenary} if for any pair of prime ideals
-$\mathfrak p \subset \mathfrak q$, all maximal chains of primes
+$\mathfrak p \subset \mathfrak q$, there exists an integer bounding the
+lengths of all finite chains of prime ideals
 $\mathfrak p = \mathfrak p_0 \subset \mathfrak p_1 \subset \ldots \subset
-\mathfrak p_e = \mathfrak q$ have the same (finite) length.
+\mathfrak p_e = \mathfrak q$ and all maximal such chains have the same length.
 \end{definition}
 
 \begin{lemma}