Ideal Quotient

In abstract algebra, if I and J are ideals of a commutative ring R, their ideal quotient (I : J) is the set

Then (I : J) is itself an ideal in R. The ideal quotient is viewed as a quotient because if and only if . The ideal quotient is useful for calculating primary decompositions. It also arises in the description of the set difference in algebraic geometry.

(I : J) is sometimes referred to as a colon ideal because of the notation. There is an unrelated notion of the inverse of an ideal, known as a fractional ideal which is defined for Dedekind rings.