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  "body": "\n## Summary\n\nA Prüfer domain is an [[integral domain|09S3PFBEPM]] in which every nonzero finitely generated ideal is [[invertible|E190TAV2S6]].\nEquivalently, its [[localization|4C1AE84N5N]] at every prime ideal is a valuation ring.\nPrüfer domains are the non-Noetherian analogue of Dedekind domains: a Noetherian Prüfer domain is exactly a Dedekind domain.\nEvery [[Bézout domain|HGEX40P0Z0]] is a Prüfer domain.\n\n## Explanations\n\n### L1 human\n\nIn a Dedekind domain, such as the ring of integers of a number field, every nonzero ideal can be \"divided out\": it has an inverse fractional ideal.\nA Prüfer domain keeps this property for finitely generated ideals, without asking that every ideal be finitely generated.\nExamples are $\\Z$, every Dedekind domain, every valuation ring, and the ring of entire functions on $\\C$.\nThe polynomial ring $\\Z[x]$ is not Prüfer: its localization at the prime $(2, x)$ contains neither $2/x$ nor $x/2$, so it is not a valuation ring.\n",
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