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  "body": "\n## Statement\n\nLet $R$ be a commutative [[Noetherian ring|0MVH4Y1CRZ]] and let $I = (x_1, \\dots, x_n)$ be an ideal of $R$ [[generated|030TZ0DQWT]] by $n$ elements.\nLet $P$ be a [[prime ideal|06HC6HK0Z8]] of $R$ that is [[minimal|S2D8823P8B]] among the prime ideals containing $I$.\nThen the [[height|9BKCZPMQT6]] of $P$ is at most $n$.\n\n## Remarks\n\nThe proof is by induction on $n$, with [[Krull's principal ideal theorem|SJZW2HN1SC]] as the case $n = 1$.\nFor the step, take a prime $Q \\subsetneq P$ with no prime strictly between them.\nSince $P$ is minimal over $I$, some $x_i$ lies outside $Q$, and one shows that $Q$ is minimal over an ideal with $n - 1$ generators.\nFor example, in $k[x, y, z]$ over a field $k$, the prime $(x, y)$ is minimal over the ideal generated by $x$ and $y$, and its height is $2$.\nA converse holds: every prime ideal of height $n$ in a Noetherian ring is minimal over some ideal generated by $n$ elements.\nAs a consequence, every prime ideal of a Noetherian ring has finite height.\nA Noetherian local ring has finite [[Krull dimension|MVGBS3J1P7]], at most the number of generators of its maximal ideal.\n\n## Explanations\n\n### L1 human\n\nCutting a space by $n$ equations lowers its dimension by at most $n$.\nIn algebra: a prime that is minimal over an ideal with $n$ generators has height at most $n$.\n\n### L2 human\n\nThe theorem is the base of dimension theory of Noetherian local rings.\nIt gives $\\dim R \\le$ the minimal number of generators of the maximal ideal, which leads to [[regular local rings|63XXTF6B5M]] and to [[systems of parameters|95QTEPMDA4]].\nIt also shows that the prime ideals of a Noetherian ring satisfy the descending chain condition.\n",
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