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  "body": "\n## Proof\n\n1. Let $R$ be an [[integral domain|09S3PFBEPM]], and let $a, b, c \\in R$ with $a \\neq 0$ and $a b = a c$.\n2. Subtract $a c$ from both sides. This gives $a b - a c = 0$.\n3. By the distributive law of the [[ring|0894J980RK]] $R$, $a (b - c) + a c = a (b - c + c) = a b$. So $a (b - c) = a b - a c = 0$.\n4. In an integral domain, a product is $0$ only if one factor is $0$. Since $a \\neq 0$, we get $b - c = 0$.\n5. Adding $c$ to both sides gives $b = c$.\n\n## Idea\n\nMove everything to one side and factor out $a$. The product $a (b - c)$ is zero, and an integral domain has no zero divisors. So the factor $b - c$ must vanish, even though $a$ need not have an inverse.\n",
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        "locator": "Properties",
        "url": "https://en.wikipedia.org/wiki/Integral_domain"
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  "msc": [
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  "name": "Proof of cancellation in an integral domain",
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