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  "body": "\n## Statement\n\nLet $R$ be a [[Noetherian|0670T59HGE]] commutative ring and let $S \\subseteq R$ be a [[multiplicative set|AN9SKEZ3VY]].\nThen the [[localization|4C1AE84N5N]] $S^{-1}R$ is a Noetherian ring.\n\n## Remarks\n\nLet $\\lambda \\colon R \\to S^{-1}R$ be the canonical map.\nEach ideal $J$ of $S^{-1}R$ is the [[extension|5AFH94P4S3]] of its contraction: $J = S^{-1}\\lambda^{-1}(J)$.\nIndeed, if $a/s \\in J$, then $a/1 = s \\cdot (a/s) \\in J$, so $a \\in \\lambda^{-1}(J)$.\nHence a strictly ascending chain of ideals in $S^{-1}R$ gives a strictly ascending chain of their contractions in $R$, which cannot be infinite.\nThe same argument shows that $S^{-1}M$ is a Noetherian $S^{-1}R$-module when $M$ is a Noetherian $R$-module.\n\nExample: $\\Z_{(p)}$, $\\Z[1/n]$, $\\Q$ and the local rings $k[x_1, \\ldots, x_n]_{\\mathfrak p}$ of a polynomial ring over a field $k$ are Noetherian.\n\nThe converse fails: Noetherianity is not a [[local property|3S58E70Z7Z]].\nAn infinite product $R = \\prod_{i \\in \\N} k$ of copies of a field $k$ is not Noetherian, but each localization $R_{\\mathfrak p}$ at a prime ideal is a field.\n\n## Explanations\n\n### L1 human\n\nA ring is Noetherian when it has no infinite strictly increasing chain of ideals.\nAllowing new denominators never creates such a chain, because every ideal of the new ring comes from an ideal of the old ring.\nSo the good finiteness property of the old ring passes to the new one.\n\n### L2 human\n\nThis fact is used all the time with $S = R \\setminus \\mathfrak p$: the local rings $R_{\\mathfrak p}$ of a Noetherian ring are Noetherian local rings.\nThus the theory of Noetherian local rings, such as [[Krull dimension|MVGBS3J1P7]], the [[embedding dimension|DP3KSCRKFP]] and [[regular local rings|63XXTF6B5M]], applies to every prime of a Noetherian ring.\nTogether with [[Hilbert's basis theorem|0YKA9J3XEE]] and passage to quotients, it shows that the coordinate rings of affine varieties and all their localizations are Noetherian.\n",
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        "url": "https://en.wikipedia.org/wiki/Localization_(commutative_algebra)"
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  "name": "Localization of a Noetherian ring is Noetherian",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/C61/C618HAK3X6.md",
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