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  "body": "\n## Statement\n\nLet $R$ be an [[integral domain|09S3PFBEPM]] and let $a, b, c \\in R$ with $a \\neq 0$.\nIf $a b = a c$, then $b = c$.\n\n## Remarks\n\nThe proof is short: $a b = a c$ gives $a (b - c) = 0$, and $a \\neq 0$ gives $b - c = 0$.\nThe converse also holds: a nonzero commutative ring with this cancellation property is an integral domain.\nCancellation fails in $\\mathbb{Z}/6\\mathbb{Z}$: $2 \\cdot 1 = 2 \\cdot 4$, but $1 \\neq 4$.\n\n## Explanations\n\n### L1 human\n\nIn the integers, you can cancel a nonzero factor from both sides of an equation.\nFrom $3 b = 3 c$ you get $b = c$.\nThis works in every integral domain, even when you cannot divide by $a$.\n",
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