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  "body": "\n## Definition\n\nLet $R$ be a nonzero [[commutative ring|0GYGHKX834]].\nA chain of [[prime ideals|06HC6HK0Z8]] $P_0 \\subsetneq P_1 \\subsetneq \\cdots \\subsetneq P_n$ of $R$ has length $n$.\nThe *Krull dimension* $\\dim R$ is the supremum of the lengths of all such chains, an element of $\\set{0, 1, 2, \\dots} \\cup \\set{\\infty}$.\n\n## Remarks\n\nFor the zero ring, which has no prime ideals, the dimension is set to $-1$ or $-\\infty$ by convention.\nExample: $\\dim \\Z = 1$; the chain $(0) \\subsetneq (p)$ for a prime $p$ has length $1$, and no chain is longer, since every nonzero prime ideal of $\\Z$ is maximal.\nNon-example of a chain: $(0) \\subsetneq (6) \\subsetneq (3)$ does not show $\\dim \\Z \\ge 2$, because $(6)$ is not a prime ideal.\nFor a field $k$, $\\dim k[x_1, \\dots, x_n] = n$, and the polynomial ring $k[x_1, x_2, \\dots]$ in infinitely many variables has infinite dimension.\nA [[Noetherian ring|0MVH4Y1CRZ]] can still have infinite dimension, by [[an example of Nagata|YEJVKCY78E]].\n\n## Explanations\n\n### L1 human\n\nThe Krull dimension of a ring is the length of the longest strictly increasing chain of prime ideals.\n",
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