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.
Read more about Ideal Quotient: Properties, Calculating The Quotient, Geometric Interpretation
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