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  "body": "\n## Definition\n\nLet $R$ be an [[integral domain|09S3PFBEPM]] with [[field of fractions|0VF11ZTTMD]] $K$.\n$R$ is a *Prüfer domain* if every nonzero finitely generated [[ideal|0MYSA6MYNA]] $I$ of $R$ is [[invertible|E190TAV2S6]].\nThat is, there is an $R$-submodule $J$ of $K$ with $IJ = R$.\n\n## Remarks\n\nEquivalent form: for every prime ideal $P$ of $R$, the localization $R_P$ is a valuation ring, that is, for every nonzero $x \\in K$, $x \\in R_P$ or $x^{-1} \\in R_P$.\nExample: every [[principal ideal domain|0RZYDVZ475]] is Prüfer; a nonzero ideal $aR$ has inverse $a^{-1}R$.\nNon-example: in $\\Z[x]$, the ideal $I = (2, x)$ is not invertible; the only $y \\in \\Q(x)$ with $yI \\subseteq \\Z[x]$ are the elements of $\\Z[x]$, so any $J$ with $IJ \\subseteq \\Z[x]$ satisfies $IJ \\subseteq I \\ne \\Z[x]$.\nA Noetherian Prüfer domain is a Dedekind domain; a Prüfer domain that is a [[GCD domain|ACDEK1GD9G]] is a [[Bézout domain|HGEX40P0Z0]].\nPrüfer domains are integrally closed.\n",
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