Universally catenary ring

Remarks

An equivalent condition is that the polynomial ring $R[X_1, \dots, X_n]$ is catenary for every $n \ge 0$. This holds because every finitely generated $R$-algebra is a quotient of such a ring, and a quotient of a catenary ring is catenary. Every Cohen–Macaulay ring is universally catenary; for example a field $k$ is, so $k[x, y]/(xy)$ is catenary. Nagata gave a Noetherian local domain that is catenary but not universally catenary: it satisfies the condition for $n = 0$ and fails it for some polynomial ring over it.