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  "body": "\n## Statement\n\nLet $A \\subseteq B$ be commutative rings with $B$ [[integral over|B7X0KFY1S1]] $A$.\nLet $\\mathfrak{p}$ be a [[prime ideal|00MHHZ9SNE]] of $A$.\nThen there is a prime ideal $\\mathfrak{q}$ of $B$ with $\\mathfrak{q} \\cap A = \\mathfrak{p}$.\n\n## Remarks\n\nOne [[localizes|4C1AE84N5N]] at $\\mathfrak{p}$: the ring $B_{\\mathfrak{p}}$ is integral over the [[local ring|MZ02BKPHJA]] $A_{\\mathfrak{p}}$ and is not zero, since $A_{\\mathfrak{p}} \\subseteq B_{\\mathfrak{p}}$.\nTake a maximal ideal $\\mathfrak{n}$ of $B_{\\mathfrak{p}}$.\nThen $B_{\\mathfrak{p}}/\\mathfrak{n}$ is a field integral over $A_{\\mathfrak{p}}/(\\mathfrak{n} \\cap A_{\\mathfrak{p}})$, so the latter is a field by [[integral extensions of fields|5GCANV7337]], and $\\mathfrak{n} \\cap A_{\\mathfrak{p}}$ is the maximal ideal $\\mathfrak{p} A_{\\mathfrak{p}}$.\nThe [[contraction|5AFH94P4S3]] of $\\mathfrak{n}$ to $B$ is the wanted $\\mathfrak{q}$.\n\nExample: for $\\Z \\subseteq \\Z[i]$ and $\\mathfrak{p} = 5\\Z$, the prime $\\mathfrak{q} = (2 + i)$ satisfies $\\mathfrak{q} \\cap \\Z = 5\\Z$, since $5 = (2 + i)(2 - i)$.\n\nIntegrality is needed: for $\\Z \\subseteq \\Q$, the only prime ideal of $\\Q$ is $0$, so no prime of $\\Q$ lies over $2\\Z$.\nInjectivity is needed too: for the integral map $\\Z \\to \\Z/2\\Z$, no prime lies over $3\\Z$.\n\n## Explanations\n\n### L1 human\n\nThe theorem says that every prime ideal of the small ring is the trace of a prime ideal of the big ring.\nIn geometry: the map of spaces given by an integral extension is surjective, every point downstairs has a point above it.\n\n### L2 human\n\nLying over is the first of the Cohen–Seidenberg theorems; [[going-up|GHKYR5PBQC]], [[incomparability|Q2XBHJ96GV]] and [[going-down|YE52MQ13MV]] build on it.\nIn number theory it says that each prime number $p$ is divisible by some prime ideal of the ring of integers of a number field.\nTogether with going-up and incomparability it gives $\\dim A = \\dim B$ for an integral extension.\n",
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