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  "body": "\n## Definition\n\nA *discrete valuation ring* is a [[principal ideal domain|0CMD0BCH8T]] $R$ that has exactly one nonzero [[prime ideal|00MHHZ9SNE]].\n\n## Remarks\n\nEquivalently, $R$ is an [[integral domain|0QVD7WBFGN]] with [[field of fractions|0F2PTGX1A2]] $K$ such that there is a surjective [[discrete valuation|A2KKZAV29W]] $v \\colon K^{\\times} \\to \\Z$ with $R = \\set{x \\in K^{\\times} : v(x) \\ge 0} \\cup \\set{0}$.\nA generator $\\pi$ of the maximal ideal is a uniformizer; every nonzero $x \\in K$ is $x = u \\pi^{v(x)}$ with $u$ a unit of $R$.\nThe nonzero ideals of $R$ are exactly $(\\pi^n)$ for $n \\ge 0$.\n\nExample: for a prime $p$, the ring $\\Z_{(p)} = \\set{a/b : a, b \\in \\Z,\\ p \\nmid b}$ is a discrete valuation ring.\nEvery nonzero element is $p^n u$ with $u$ a unit and $n \\ge 0$, so every nonzero ideal is $(p^n)$, which is principal.\nThe only nonzero prime among these is $(p)$.\nThe ring $k[\\![t]\\!]$ of formal power series over a field $k$ and the ring $\\Z_p$ of $p$-adic integers are other examples.\n\nNon-example: $\\Z$ is a principal ideal domain, but it has infinitely many nonzero prime ideals $(2), (3), (5), \\ldots$, so it is not a discrete valuation ring.\nA field is not a discrete valuation ring either: it has no nonzero prime ideal.\n",
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