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  "body": "\n## Statement\n\nLet $R$ be a [[Euclidean domain|0GJ2HRG95T]].\nThen $R$ is a [[principal ideal domain|0RZYDVZ475]].\n\n## Remarks\n\nThe proof: let $I \\neq 0$ be an ideal and let $b \\in I$ be a nonzero element with $\\delta(b)$ minimal. For $a \\in I$, write $a = b q + r$. Then $r \\in I$ and $\\delta(r) \u003c \\delta(b)$ is impossible, so $r = 0$ and $I = (b)$.\nThe converse is false: $\\mathbb{Z}[\\tfrac{1 + \\sqrt{-19}}{2}]$ is a principal ideal domain that is not Euclidean.\n\n## Explanations\n\n### L1 human\n\nTake a nonzero element of the ideal that is as small as possible.\nEvery other element of the ideal leaves remainder zero when you divide it by this one.\nSo the smallest element generates the ideal.\n",
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