---
tag: 16J9DM8V57
kind: concept
name: Ideal quotient
msc: [13A15]
sources:
  - url: https://en.wikipedia.org/wiki/Ideal_quotient
same_as:
  wikidata: Q17098054
  wikipedia: Ideal quotient
---

## Summary

The ideal quotient $(I : J)$ of two ideals of a commutative ring is the set of elements that multiply $J$ into $I$.
It is an ideal that contains $I$.
It plays the role of division for ideals.
It is used in primary decomposition, in the study of [[associated primes|ERFZ6S8X0X]], and in computing the closure of a difference of two algebraic sets.

## Explanations

### L1 human

The ideal quotient asks: which elements $r$ make $rJ$ land inside $I$?
In $\Z$, the quotient $(12\Z : 4\Z)$ is $3\Z$, because $4r$ is a multiple of $12$ exactly when $r$ is a multiple of $3$.
So the quotient of ideals mirrors the division $12 / 4 = 3$.
In 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.