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  "body": "\n## Summary\n\nA Bézout domain is an [[integral domain|09S3PFBEPM]] in which every finitely generated ideal is [[principal|00A2EDFT9Q]].\nIn such a ring [[Bézout's identity|0DQVHMYAGA]] holds: a gcd of $a$ and $b$ can be written as $ax + by$.\nEvery [[principal ideal domain|0RZYDVZ475]] is a Bézout domain, and a Noetherian Bézout domain is a principal ideal domain.\nBézout domains are [[GCD domains|ACDEK1GD9G]] and [[Prüfer domains|2RDAA4PCK8]].\n\n## Explanations\n\n### L1 human\n\nIn the integers, $\\gcd(a, b)$ can always be written as $ax + by$ for some integers $x, y$.\nA Bézout domain is a ring where this works for every pair of elements, even if the ring has ideals that are not finitely generated.\nThe ring of entire functions on $\\C$ and the ring of all algebraic integers are Bézout domains that are not principal ideal domains.\nThe polynomial ring $\\Z[x]$ is not Bézout: the ideal generated by $2$ and $x$ is not principal.\n",
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