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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0310REJJY8]] and let $\\mathfrak p$ be a [[prime ideal|00MHHZ9SNE]] of $R$.\nThe set $S = R \\setminus \\mathfrak p$ is a [[multiplicative set|AN9SKEZ3VY]].\nThe *localization of $R$ at $\\mathfrak p$* is the [[localization|4C1AE84N5N]]\n$$R_{\\mathfrak p} = S^{-1}R = \\set{a/s : a \\in R, \\ s \\in R \\setminus \\mathfrak p}.$$\n\n## Remarks\n\nThe ring $R_{\\mathfrak p}$ is a [[local ring|MZ02BKPHJA]].\nIts unique maximal ideal is $\\mathfrak p R_{\\mathfrak p} = \\set{a/s : a \\in \\mathfrak p, \\ s \\notin \\mathfrak p}$.\nIndeed, an element $a/s$ with $a \\notin \\mathfrak p$ has the inverse $s/a$, so every element outside $\\mathfrak p R_{\\mathfrak p}$ is a unit.\nThe [[residue field|NJ3APQ8BHH]] $R_{\\mathfrak p} / \\mathfrak p R_{\\mathfrak p}$ is called the residue field of $R$ at $\\mathfrak p$.\n\nExample: for $R = \\Z$ and $\\mathfrak p = p\\Z$ with $p$ prime, $\\Z_{(p)}$ is the ring of rationals $a/b$ in lowest terms with $p \\nmid b$.\nIts maximal ideal is $p \\Z_{(p)}$ and its residue field is $\\Z/p\\Z$.\nFor $\\mathfrak p = (0)$ in an [[integral domain|0QVD7WBFGN]] $R$, the localization $R_{(0)}$ is the [[field of fractions|0F2PTGX1A2]] of $R$.\n\nNon-example: the construction needs $\\mathfrak p$ to be prime.\nFor the ideal $6\\Z$ of $\\Z$, the complement $\\Z \\setminus 6\\Z$ contains $2$ and $3$ but not $6$, so it is not a multiplicative set.\n\nNotation: for an element $f \\in R$, the ring $R_f$ is the localization at the powers of $f$, which differs from $R_{\\mathfrak p}$.\n",
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