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  "body": "\n## Definition\n\nLet $R$ be an [[integral domain|09S3PFBEPM]] with [[field of fractions|0VF11ZTTMD]] $K$.\nThe ring $R$ is a *valuation ring* if for every nonzero $x \\in K$, at least one of $x$ and $x^{-1}$ lies in $R$.\n\n## Remarks\n\nExample: for a prime $p$, the ring $\\Z_{(p)} = \\set{a/b : a, b \\in \\Z,\\ p \\nmid b}$ is a valuation ring.\nEach nonzero rational number is $x = p^n a/b$ with $n \\in \\Z$ and $a, b$ prime to $p$; if $n \\ge 0$ then $x \\in \\Z_{(p)}$, and if $n \u003c 0$ then $x^{-1} = p^{-n} b/a \\in \\Z_{(p)}$.\nNon-example: $\\Z$ is not a valuation ring, since neither $2/3$ nor $3/2$ is an integer.\nEvery field is a valuation ring of itself.\nIf $v$ is a [[valuation|B85T5VKR3M]] on a field $K$, then $\\set{x \\in K : v(x) \\ge 0}$ is a valuation ring with fraction field $K$.\nConversely, let $R$ be a valuation ring with fraction field $K$.\nThen $K^\\times / R^\\times$, ordered by $xR^\\times \\le yR^\\times$ when $y/x \\in R$, is a totally ordered group.\nThe quotient map $K^\\times \\to K^\\times/R^\\times$ extends to a valuation whose ring is $R$.\nA valuation ring is a local ring, with maximal ideal the set of non-units, and any two of its ideals are comparable by inclusion.\n",
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