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  "body": "\n## Summary\n\nThe localization of a [[commutative ring|0310REJJY8]] $R$ at a [[prime ideal|00MHHZ9SNE]] $\\mathfrak p$ is the ring $R_{\\mathfrak p}$ of fractions whose denominators lie outside $\\mathfrak p$.\nIt is a [[local ring|MZ02BKPHJA]] with maximal ideal $\\mathfrak p R_{\\mathfrak p}$.\nIt keeps the information about $R$ near the point $\\mathfrak p$ of the [[spectrum|FB45BHHA2J]] and forgets the rest.\nMany questions in commutative algebra reduce to questions about these local rings.\n\n## Explanations\n\n### L1 human\n\nPick a prime number $p$ and allow as denominators all integers not divisible by $p$.\nThe result $\\Z_{(p)}$ contains $1/3$ and $5/7$ when $p = 2$, but not $1/2$.\nIn this ring every prime other than $p$ has become invertible, so only divisibility by $p$ still matters.\nThis is what \"local at $p$\" means: the ring sees only the prime $p$.\nThe [[definition|N37SPDWYGS]] does the same for any prime ideal of any commutative ring.\n",
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