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  "body": "\n## Definition\n\nLet $R$ be a commutative [[Noetherian ring|0670T59HGE]].\nThe ring $R$ is *universally catenary* if every [[finitely generated|AC5TFFQT9M]] commutative $R$-algebra is [[catenary|B2CYR292J6]].\n\n## Remarks\n\nAn equivalent condition is that the [[polynomial ring|0SDC934R52]] $R[X_1, \\dots, X_n]$ is catenary for every $n \\ge 0$.\nThis holds because every finitely generated $R$-algebra is a quotient of such a ring, and a quotient of a catenary ring is catenary.\nEvery Cohen–Macaulay ring is universally catenary; for example a field $k$ is, so $k[x, y]/(xy)$ is catenary.\nNagata 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.\n",
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