Let $R$ be a commutative ring with identity. Then $R$ is $\textit{catenary}$ if for each pair of prime ideal $p \subsetneq q$, all maximal chains of prime ideals $p = p_0 \subsetneq p_1 \subsetneq \dots \subsetneq p_n = q$ have the same length.
We know that factor ring of a catenary domain is catenary. I was thinking about (somehow) a dual of this proposition: if we imbed a non-catenary domain in another domain, is it necessarily non-catenary? this trivially false; take quotient field. So I add an assumption: what if imbedding is finite? So the question is:
Let $R$ be a non-catenary domain and $f: R \to S$ be a finite monomorphism (as $R$-algebra). Can $S$ be catenary?
thank you.
