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  "body": "\n## Definition\n\nLet $(P, \\le)$ be a [[partially ordered set|0QV2JS9RJ3]].\nA finite [[chain|0S8FE7ZMFM]] $x_0 \u003c x_1 \u003c \\cdots \u003c x_n$ in $P$ has length $n$.\nThe *Krull dimension* $\\dim P$ of a nonempty $P$ is the supremum of the lengths of the finite chains in $P$, an element of $\\N \\cup \\set{\\infty}$.\nThe empty poset has Krull dimension $-\\infty$ by convention; some authors use $-1$ instead.\n\n## Remarks\n\nExample: the power set of $\\set{1, \\ldots, n}$ ordered by inclusion has Krull dimension $n$.\nThe chain $\\varnothing \\subset \\set{1} \\subset \\cdots \\subset \\set{1, \\ldots, n}$ has length $n$, and a longer chain would need two subsets of the same size.\nExample: $\\N$ with its usual order has Krull dimension $\\infty$.\nThe dimension can be infinite even when every chain is finite: the disjoint union of chains of lengths $1, 2, 3, \\ldots$ has no infinite chain but has dimension $\\infty$.\nFor a commutative ring $R$, the [[Krull dimension of the ring|MVGBS3J1P7]] is the Krull dimension of its set of prime ideals ordered by inclusion.\n",
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