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  "body": "\n## Statement\n\nLet $R$ be a [[ring|0894J980RK]], let $S$ be a [[subring|0859D58WPD]] of $R$ and let $I$ be a two-sided ideal of $R$.\nThen $S + I = \\{s + x : s \\in S, x \\in I\\}$ is a subring of $R$, and $I$ is a two-sided ideal of $S + I$.\nThe intersection $S \\cap I$ is a two-sided ideal of $S$.\nThe map $S/(S \\cap I) \\to (S + I)/I$, $s + (S \\cap I) \\mapsto s + I$, is a [[ring isomorphism|0B6QC0C0G1]].\n\n## Remarks\n\nThe proof applies the [[first isomorphism theorem|0ZMYX7GMH0]] to the map $S \\to (S + I)/I$, $s \\mapsto s + I$. It is surjective, and its kernel is $S \\cap I$.\nThe theorem is also called the diamond theorem, after the picture of the four rings $S \\cap I$, $S$, $I$ and $S + I$.\n\n## Explanations\n\n### L1 human\n\nThe theorem compares two quotients.\nDivide the subring $S$ by its part inside $I$, or divide the larger ring $S + I$ by $I$.\nThe two quotients are the same ring.\n",
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