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  "body": "\n## Statement\n\nLet $R$ be a [[commutative ring|0GYGHKX834]] and let $P$ be an ideal of $R$.\nThen the [[quotient ring|0VFW4Y69BT]] $R/P$ is an [[integral domain|09S3PFBEPM]] if and only if $P$ is a [[prime ideal|06HC6HK0Z8]].\n\n## Remarks\n\nThe condition $1 + P \\neq 0 + P$ in $R/P$ is the condition $P \\neq R$.\nThe condition that $(a + P)(b + P) = 0$ implies $a + P = 0$ or $b + P = 0$ is the condition that $a b \\in P$ implies $a \\in P$ or $b \\in P$.\nFor example, $\\mathbb{Z}/n\\mathbb{Z}$ is an integral domain exactly when $n$ is prime or $n = 0$.\n\n## Explanations\n\n### L1 human\n\nThe theorem translates a property of the ideal into a property of the quotient ring.\nAn element is zero in $R/P$ exactly when it lies in $P$.\nSo \"a product in $P$ has a factor in $P$\" becomes \"a zero product has a zero factor\".\n",
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  "name": "R/P is an integral domain iff P is prime",
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