Catenary ring

Definition

Let $R$ be a commutative ring. A chain of prime ideals $\mathfrak p_0 \subsetneq \mathfrak p_1 \subsetneq \cdots \subsetneq \mathfrak p_n$ has length $n$. It is saturated if no prime ideal $\mathfrak q$ satisfies $\mathfrak p_{i-1} \subsetneq \mathfrak q \subsetneq \mathfrak p_i$ for some $i$. The ring $R$ is catenary if the following holds for all prime ideals $\mathfrak p \subseteq \mathfrak q$ of $R$. Every saturated chain of prime ideals from $\mathfrak p$ to $\mathfrak q$ is finite, and all such chains have the same length.