Commutative Ring Theory
《Commutative Ring Theory》 · 第 31 页 · Section 5「Chains of prime ideals」
A strictly increasing (or decreasing) chain $\mathfrak p_0,\mathfrak p_1,\ldots$ of prime ideals is said to be saturated if there do not exist prime ideals strictly contained between any two consecutive terms. We say that $A$ is a catenary ring if the following condition is satisfied: for any prime ideals $\mathfrak p$ and $\mathfrak p'$ of $A$ with $\mathfrak p\subset\mathfrak p'$, there exists a saturated chain of prime ideals starting from $\mathfrak p$ and ending at $\mathfrak p'$, and all such chains have the same (finite) length.