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Concept# Ideal quotient

Summary

In abstract algebra, if I and J are ideals of a commutative ring R, their ideal quotient (I : J) is the set
:(I : J) = {r \in R \mid rJ \subseteq I}
Then (I : J) is itself an ideal in R. The ideal quotient is viewed as a quotient because KJ \subseteq I if and only if K \subseteq (I : J). The ideal quotient is useful for calculating primary decompositions. It also arises in the description of the set difference in algebraic geometry (see below).
(I : J) is sometimes referred to as a colon ideal because of the notation. In the context of fractional ideals, there is a related notion of the inverse of a fractional ideal.
Properties
The ideal quotient satisfies the following properties:
*(I :J)=\mathrm{Ann}_R((J+I)/I) as R-modules, where \mathrm{Ann}_R(M) denotes the annihilator of M as an R-module.
*J \subseteq I \Leftrightarrow (I : J) = R (in particular,

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