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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0310REJJY8]].\nA chain of [[prime ideals|00MHHZ9SNE]] $\\mathfrak p_0 \\subsetneq \\mathfrak p_1 \\subsetneq \\cdots \\subsetneq \\mathfrak p_n$ has length $n$.\nIt is saturated if no prime ideal $\\mathfrak q$ satisfies $\\mathfrak p_{i-1} \\subsetneq \\mathfrak q \\subsetneq \\mathfrak p_i$ for some $i$.\nThe ring $R$ is *catenary* if the following holds for all prime ideals $\\mathfrak p \\subseteq \\mathfrak q$ of $R$.\nEvery saturated chain of prime ideals from $\\mathfrak p$ to $\\mathfrak q$ is finite, and all such chains have the same length.\n\n## Remarks\n\nEvery principal ideal domain is catenary.\nIts nonzero primes are maximal, so the saturated chains are $(0) \\subsetneq \\mathfrak m$ and the trivial chains, and between two given primes they all have the same length.\nNagata constructed Noetherian local domains that are not catenary: in such a ring there are primes $\\mathfrak p \\subseteq \\mathfrak q$ joined by saturated chains of two different lengths.\nMany authors include the condition that $R$ is [[Noetherian|0670T59HGE]] in the definition.\nA stronger notion is that of a [[universally catenary ring|5WAFPTH09V]].\n",
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