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  "body": "\n## Definition\n\nLet $R$ be an [[integral domain|09S3PFBEPM]].\n$R$ is a *Bézout domain* if every finitely generated [[ideal|0MYSA6MYNA]] of $R$ is [[principal|00A2EDFT9Q]].\n\n## Remarks\n\nEquivalent form: for all $a, b \\in R$ the ideal $(a, b)$ is principal; the general case follows by induction on the number of generators.\nThen $(a, b) = (d)$ for a gcd $d$ of $a$ and $b$, and $d = ax + by$ for some $x, y \\in R$.\nExample: every [[principal ideal domain|0RZYDVZ475]], such as $\\Z$ or $\\Q[x]$, is a Bézout domain.\nNon-example: in $\\Z[x]$ the ideal $(2, x)$ is not principal.\nA generator would divide $2$, so it would be $\\pm 1$ or $\\pm 2$.\nBut $(2, x) \\ne \\Z[x]$, because its elements have even constant term, and $2$ does not divide $x$.\nA Bézout domain is a [[GCD domain|ACDEK1GD9G]]; a Noetherian Bézout domain is a principal ideal domain.\n",
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