Let $R$ be a commutative ring. The following are well-known:

  1. If $R$ is catenary and $\mathfrak{a}\subseteq R$ is an ideal, then $R/\mathfrak{a}$ is catenary.
  2. If $R$ is catenary and $S\subseteq R$ is a subset, then $S^{-1}R$ is catenary.

This means that catenarity is preserved along two special kinds of epimorphisms of commutative rings. In view of this question (and its answers) one may wonder:

Suppose that $R$ is catenary, and let $f\colon R\twoheadrightarrow R'$ be an epimorphism of commutative rings. Is $R'$ catenary?



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