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

(I : J) = \{r \in R | rJ \subset I\}

Then (I : J) is itself an ideal in R. The ideal quotient is viewed as a quotient because IJ \subset K if and only if I \subset K : J. 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:

  • (I :J)=\mathrm{Ann}_R((J+I)/I) as R-modules, where \mathrm{Ann}_R(M) denotes the annihilator of M as an R-module.
  • J \subset I \Rightarrow I : J = R
  • I : R = I
  • R : I = R
  • I : (J + K) = (I : J) \cap (I : K)
  • I : (r) = \frac{1}{r}(I \cap (r)) (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

I : J = (I : (g_1)) \cap (I : (g_2)) = \left(\frac{1}{g_1}(I \cap (g_1))\right) \cap \left(\frac{1}{g_2}(I \cap (g_2))\right)

Then elimination theory can be used to calculate the intersection of I with (g1) and (g2):

I \cap (g_1) = tI + (1-t)(g_1) \cap k[x_1, \dots, x_n], \quad I \cap (g_2) = tI + (1-t)(g_1) \cap k[x_1, \dots, x_n]

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 I \cap (g_1).

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
I(V) : I(W) = I(V \setminus W)

where I(\bullet) 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
Z(I : J) = \mathrm{cl}(Z(I) \setminus Z(J))

where \mathrm{cl}(\bullet) denotes the Zariski closure, and Z(\bullet) 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:

Z(I : J^{\infty}) = \mathrm{cl}(Z(I) \setminus Z(J))

where J^\infty = J + J^2 + \cdots + J^n + \cdots.

References[edit]

  1. ^ David Cox, John Little, and 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

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.