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  "body": "\n## Summary\n\nThe ideal quotient $(I : J)$ of two ideals of a commutative ring is the set of elements that multiply $J$ into $I$.\nIt is an ideal that contains $I$.\nIt plays the role of division for ideals.\nIt is used in primary decomposition, in the study of [[associated primes|ERFZ6S8X0X]], and in computing the closure of a difference of two algebraic sets.\n\n## Explanations\n\n### L1 human\n\nThe ideal quotient asks: which elements $r$ make $rJ$ land inside $I$?\nIn $\\Z$, the quotient $(12\\Z : 4\\Z)$ is $3\\Z$, because $4r$ is a multiple of $12$ exactly when $r$ is a multiple of $3$.\nSo the quotient of ideals mirrors the division $12 / 4 = 3$.\nIn geometry, the ideal quotient corresponds to removing the zero set of $J$ from the zero set of $I$ and taking what is left, up to closure.\n",
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