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?



Are integral extensions of a catenary ring still catenary?
No. Nagata's famous family of examples of non-catenary rings yields a non-catenary finite extension of a catenary noetherian local domain.


**Reference:** M. Nagata, [*On the chain problem of prime ideals*](https://projecteuclid.org/euclid.nmj/1118799769), Nagoya Math. J. 10 (1956), 51-64.


