{
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  "body": "\n## Statement\n\nLet $R$ be a [[ring|0894J980RK]], let $I$ be a two-sided ideal of $R$ and let $\\pi \\colon R \\to R/I$ be the quotient map.\nLet $\\mathcal{A}$ be the set of the left [[ideals|0MYSA6MYNA]] $J$ of $R$ with $I \\subseteq J$.\nLet $\\mathcal{B}$ be the set of the left ideals of $R/I$.\nThe map $\\mathcal{A} \\to \\mathcal{B}$, $J \\mapsto J/I = \\pi(J)$, is a bijection.\nIts inverse is $K \\mapsto \\pi^{-1}(K)$.\nThis bijection and its inverse preserve inclusion.\n\n## Remarks\n\nThe same bijection holds for right ideals, for two-sided ideals, and for subrings that contain $I$.\nFor two-sided ideals $J \\supseteq I$, the [[third isomorphism theorem|073C80EC63]] gives $(R/I)/(J/I) \\cong R/J$.\nThe ideals of $\\mathbb{Z}/12\\mathbb{Z}$ correspond to the ideals $d\\mathbb{Z}$ with $d \\mid 12$, so there are six of them.\n\n## Explanations\n\n### L1 human\n\nThe ideals of a quotient ring $R/I$ are the ideals of $R$ that contain $I$, seen in the quotient.\nSo you can understand the ideals of $R/I$ from the ideals of $R$.\nFor example, the ideals of $\\mathbb{Z}/12\\mathbb{Z}$ come from the divisors $1, 2, 3, 4, 6, 12$ of $12$.\n",
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