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  "body": "\n## Statement\n\nLet $R$ be a [[commutative ring|0GYGHKX834]] and let $I_1, \\dots, I_n$ be ideals of $R$, with $n \\ge 1$.\nSuppose that $I_j$ and $I_k$ are [[coprime|0MDJP82TEH]] for all $j \\neq k$.\nThen the map\n$$R/(I_1 \\cap \\dots \\cap I_n) \\to R/I_1 \\times \\dots \\times R/I_n, \\qquad a + (I_1 \\cap \\dots \\cap I_n) \\mapsto (a + I_1, \\dots, a + I_n)$$\nis a [[ring isomorphism|0B6QC0C0G1]].\n\n## Remarks\n\nUnder the same hypothesis, $I_1 \\cap \\dots \\cap I_n = I_1 I_2 \\cdots I_n$.\nFor $R = \\mathbb{Z}$ and pairwise coprime integers $m_1, \\dots, m_n$, the theorem gives $\\mathbb{Z}/(m_1 \\cdots m_n)\\mathbb{Z} \\cong \\mathbb{Z}/m_1\\mathbb{Z} \\times \\dots \\times \\mathbb{Z}/m_n\\mathbb{Z}$.\nFor example, $\\mathbb{Z}/6\\mathbb{Z} \\cong \\mathbb{Z}/2\\mathbb{Z} \\times \\mathbb{Z}/3\\mathbb{Z}$.\n\n## Explanations\n\n### L1 human\n\nFor integers, the theorem says that a number modulo $15$ is the same information as the number modulo $3$ and modulo $5$.\nFor each pair of remainders, exactly one class modulo $15$ has them.\nThe ring version says the same for any family of pairwise coprime ideals.\n",
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