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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0GYGHKX834]] and let $I$ and $J$ be ideals of $R$.\nThe ideals $I$ and $J$ are coprime if $I + J = R$, where $I + J$ is the [[sum of the ideals|0MBJVDSK1R]].\nEquivalently, there are $a \\in I$ and $b \\in J$ with $a + b = 1$.\n\n## Remarks\n\nIn $\\mathbb{Z}$, $m\\mathbb{Z} + n\\mathbb{Z} = \\gcd(m, n)\\mathbb{Z}$. So $m\\mathbb{Z}$ and $n\\mathbb{Z}$ are coprime if and only if $\\gcd(m, n) = 1$.\nIf $I$ and $J$ are coprime, then $I \\cap J = I J$.\nTwo distinct [[maximal ideals|09J6KGT65G]] are always coprime.\n\n## Explanations\n\n### L1 human\n\nFor integers, $3$ and $5$ are coprime, and $2 \\cdot 3 - 1 \\cdot 5 = 1$.\nSo $1$ is a multiple of $3$ plus a multiple of $5$.\nCoprime ideals generalize this: $1$ is an element of $I$ plus an element of $J$.\n",
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