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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0GYGHKX834]] and let $P$ be a [[prime ideal|06HC6HK0Z8]] of $R$.\nThe *height* $\\operatorname{ht} P$ of $P$ is the supremum of the integers $n$ for which there is a chain of prime ideals $P_0 \\subsetneq P_1 \\subsetneq \\cdots \\subsetneq P_n = P$ of $R$.\n\n## Remarks\n\nThe height is an element of $\\set{0, 1, 2, \\dots} \\cup \\set{\\infty}$, and $\\operatorname{ht} P = \\dim R_P$, the [[Krull dimension|1B4G153ZGE]] of the localization at $P$.\nExample: in $k[x, y]$ over a field $k$, $\\operatorname{ht}(x, y) = 2$, because of the chain $(0) \\subsetneq (x) \\subsetneq (x, y)$, and the dimension of $k[x, y]$ is $2$.\nNon-example of a chain ending at $(x, y)$: $(x^2) \\subsetneq (x) \\subsetneq (x, y)$ does not give height $3$, because $(x^2)$ is not prime.\nThe prime ideals of height $0$ are the [[minimal prime ideals|S2D8823P8B]] of $R$.\nFor any ideal $I \\ne R$, the height of $I$ is the least height of a prime ideal that contains $I$.\nOne has $\\operatorname{ht} P + \\dim R/P \\le \\dim R$.\n\n## Explanations\n\n### L1 human\n\nThe height of a prime ideal is the length of the longest strictly increasing chain of prime ideals that ends at it.\n",
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