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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
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 (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.
The ideal quotient satisfies the following properties:
as -modules, where denotes the annihilator of as an -module.
(in particular, )
(as long as R is an integral domain)
The above properties can be used to calculate the quotient of ideals in a polynomial ring given their generators. For example, if I = (f1, f2, f3) and J = (g1, g2) are ideals in k[x1, ..., xn], then
Then elimination theory can be used to calculate the intersection of I with (g1) and (g2):
Calculate a Gröbner basis for with respect to lexicographic order. Then the basis functions which have no t in them generate .
The ideal quotient corresponds to set difference in algebraic geometry. More precisely,
If W is an affine variety (not necessarily irreducible) and V is a subset of the affine space (not necessarily a variety), then
where denotes the taking of the ideal associated to a subset.
If I and J are ideals in k[x1, ..., xn], with k an algebraically closed field and I radical then
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 .
In ,
In algebraic number theory, the ideal quotient is useful while studying fractional ideals. This is because the inverse of any invertible fractional ideal of an integral domain is given by the ideal quotient .
One geometric application of the ideal quotient is removing an irreducible component of an affine scheme.
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