A (commutative unitary) Noetherian ring $R$ of finite dimension is said to be catenary if for every prime ideal $\mathfrak{p}$ of $R$ one has $\mathrm{ht}(\mathfrak{p})+\mathrm{dim}(R/\mathfrak{p})=\mathrm{dim}R$.

Let $A$ be a catenary ring and $B$ a finite extension of $A$ (*id est* $A\subseteq B$ and $B$ is a finite $A$-module). Is $B$ a catenary ring?

catenaryequivalent to the usual one (that requires that for all primes $p \subsetneq q$ all maximal prime chains $p \subsetneq \cdots \subsetneq q$ have the same length)? $\endgroup$ – tj_ Mar 24 '19 at 10:25