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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### Famous quotes containing the word ideal:

“One who does not know how to discover the pathway to his *ideal* lives more frivolously and impudently than the man without an *ideal*.”

—Friedrich Nietzsche (1844–1900)