Ideal quotient


In abstract algebra, if I and J are ideals of a commutative ring R, their ideal quotient is the set
Then 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.
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:
The above properties can be used to calculate the quotient of ideals in a polynomial ring given their generators. For example, if I = and J = are ideals in k, then
Then elimination theory can be used to calculate the intersection of I with and :
Calculate a Gröbner basis for tI + with respect to lexicographic order. Then the basis functions which have no t in them generate.

Geometric interpretation

The ideal quotient corresponds to set difference in algebraic geometry. More precisely,
where denotes the taking of the ideal associated to a subset.
where denotes the Zariski closure, and denotes the taking of the variety defined by an ideal.
If I is not radical, then the same property holds if we saturate the ideal J:
where.

Examples