Ideal quotient

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

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

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, ..., x_n], 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

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, ..., x_n], 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 $(I : J^\infty )= \cup_{n \geq 1} (I:J^n)$.

References

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.