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  "body": "\n## Summary\n\nA catenary ring is a [[commutative ring|0310REJJY8]] in which, between any two nested [[prime ideals|00MHHZ9SNE]], all maximal chains of primes have the same finite length.\nIn such a ring the distance between two primes is well defined.\nMost rings met in practice are catenary, such as finitely generated algebras over a field and Cohen–Macaulay rings.\nNagata showed that some Noetherian local domains are not catenary.\n\n## Explanations\n\n### L1 human\n\nThink of the prime ideals as floors of a building, ordered by inclusion.\nA chain of primes is a staircase, and a maximal chain is one with no floor skipped.\nThe ring is catenary when every staircase between two given floors has the same number of steps.\nFor example, in $\\Z$ the only chains from $(0)$ to a prime $(p)$ are $(0) \\subsetneq (p)$, all of length $1$.\nThe [[definition|K6PPGRT5E1]] makes this precise.\n",
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