Ideal quotient


In abstract algebra, if and are ideals of a commutative ring, their ideal quotient is the set
Then is itself an ideal in. 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:
  • as -modules, where denotes the annihilator of as an -module.

  • Calculating the quotient

The above properties can be used to calculate the quotient of ideals in a polynomial ring given their generators. For example, if and are ideals in, then
Then elimination theory can be used to calculate the intersection of with and :
Calculate a Gröbner basis for 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,
  • If is an affine variety and is a subset of the affine space, then
  • If and are ideals in, with an algebraically closed field and radical then

    Examples

  • In we have.
  • 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. For example, let in be the ideals corresponding to the union of the x,y, and z-planes and x and y planes in. Then, the ideal quotient is the ideal of the z-plane in. This shows how the ideal quotient can be used to "delete" irreducible subschemes.
  • A useful scheme theoretic example is taking the ideal quotient of a reducible ideal. For example, the ideal quotient, showing that the ideal quotient of a subscheme of some non-reduced scheme, where both have the same reduced subscheme, kills off some of the non-reduced structure.
  • We can use the previous example to find the saturation of an ideal corresponding to a projective scheme. Given a homogeneous ideal the saturation of is defined as the ideal quotient where. It is a theorem that the set of saturated ideals of contained in is in bijection with the set of projective subschemes in. This shows us that defines the same projective curve as in.