Ideal quotient

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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.

Properties[edit]

The ideal quotient satisfies the following properties:

  • as -modules, where denotes the annihilator of as an -module.
  • (as long as R is an integral domain)

Calculating the quotient[edit]

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 tI + (1-t)(g1) with respect to lexicographic order. Then the basis functions which have no t in them generate .

Geometric interpretation[edit]

The ideal quotient corresponds to set difference in algebraic geometry.[1] More precisely,

  • If W is an affine variety 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 algebraically closed 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 .

Examples[edit]

  • In ,
  • 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 .[2] This shows us that defines the same projective curve as in .

References[edit]

  1. ^ David Cox; John Little; Donal O'Shea (1997). Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. Springer. ISBN 0-387-94680-2. , p.195
  2. ^ Greuel, Gert-Martin; Pfister, Gerhard (2008). A Singular Introduction to Commutative Algebra (2nd ed.). Springer-Verlag. p. 485. ISBN 9783642442544. 

Viviana Ene, Jürgen Herzog: 'Gröbner Bases in Commutative Algebra', AMS Graduate Studies in Mathematics, Vol 130 (AMS 2012)

M.F.Atiyah, I.G.MacDonald: 'Introduction to Commutative Algebra', Addison-Wesley 1969.