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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0GYGHKX834]] and let $I$ and $J$ be [[ideals|0MYSA6MYNA]] of $R$.\nThe *ideal quotient* of $I$ by $J$ is the set\n$$(I : J) = \\set{r \\in R : rj \\in I \\text{ for all } j \\in J}.$$\n\n## Remarks\n\n$(I : J)$ is an ideal of $R$ and $I \\subseteq (I : J)$.\nExample: in $\\Z$, $(12\\Z : 4\\Z) = 3\\Z$; for $r \\in \\Z$, $4r \\in 12\\Z$ holds exactly when $3 \\mid r$, and then $rj \\in 12\\Z$ for every multiple $j$ of $4$.\nNon-example: $1 \\notin (12\\Z : 4\\Z)$, since $1 \\cdot 4 = 4 \\notin 12\\Z$.\nIf $J \\subseteq I$, then $(I : J) = R$.\nThe quotient $(0 : J)$ is the [[annihilator|HB4XDKJTW8]] of $J$.\nOther notations are $I : J$ and $(I :_R J)$; the name *colon ideal* is also used.\n",
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