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  "body": "\n## Statement\n\nLet $R$ be a commutative [[Noetherian ring|0MVH4Y1CRZ]] and let $x \\in R$.\nLet $P$ be a [[prime ideal|06HC6HK0Z8]] of $R$ that is [[minimal|S2D8823P8B]] among the prime ideals containing $x$.\nThen the [[height|9BKCZPMQT6]] of $P$ is at most $1$.\n\n## Remarks\n\nAfter localizing at $P$, one may assume that $R$ is local with maximal ideal $P$, and then $R/xR$ is Artinian.\nIf $Q \\subsetneq P$ is prime, the chain of ideals $xR + Q^{(n)}$ stabilizes, where $Q^{(n)}$ is the [[symbolic power|7VFPQVCNJ7]]; with Nakayama's lemma this gives $Q^{(n)} = Q^{(n+1)}$ for large $n$.\nThen $Q^n R_Q = 0$ in $R_Q$, so $R_Q$ is Artinian, and $Q$ has height $0$.\nFor example, in $k[x, y]$ the prime $(x)$ is minimal over $x$ and has height $1$.\nThe Noetherian hypothesis is needed.\nIn a valuation domain whose value group is $\\Z \\times \\Z$ with the lexicographic order, the maximal ideal has height $2$ but is minimal over a single element.\nIf $x$ is not a zero divisor and not a unit, every prime minimal over $x$ has height exactly $1$.\n\n## Explanations\n\n### L1 human\n\nCutting a space by one equation lowers its dimension by at most one.\nIn algebra: a prime ideal that is minimal over one element sits at most one step above a minimal prime.\n\n### L2 human\n\nThe theorem, the Hauptidealsatz, is the base case of [[Krull's height theorem|TBRTZMHFTV]] for ideals generated by $n$ elements.\nThrough that generalisation, it shows that every prime ideal of a Noetherian ring has finite height.\nIt also gives the criterion that a Noetherian domain is a unique factorisation domain exactly when every height one prime is principal.\nIt is a basic tool in dimension theory, for example in the [[dimension theorem for local rings|KW3HY77V6D]].\n",
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