Ideal quotient

In abstract algebra, if I and J are ideals of a commutative ring R, their ideal quotient (I : J) is the set


 * $$(I : J) = \{r \in R \mid rJ \subseteq I\}$$

Then (I : J) is itself an ideal in R. The ideal quotient is viewed as a quotient because $$KJ \subseteq I$$ if and only if $$K \subseteq (I : J)$$. 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.
 * $$J \subseteq I \Leftrightarrow (I : J) = R$$ (in particular, $$(I : I) = (R : I) = (I : 0) = R$$)
 * $$(I : R) = I$$
 * $$(I : (JK)) = ((I : J) : K)$$
 * $$(I : (J + K)) = (I : J) \cap (I : K)$$
 * $$((I \cap J) : K) = (I : K) \cap (J : K)$$
 * $$(I : (r)) = \frac{1}{r}(I \cap (r))$$ (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, ..., 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):
 * $$I \cap (g_1) = tI + (1-t) (g_1) \cap k[x_1, \dots, x_n], \quad I \cap (g_2) = tI + (1-t) (g_2) \cap k[x_1, \dots, x_n]$$

Calculate a Gröbner basis for $$tI+(1-t)(g_1)$$ 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. 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
 * $$I(V) : I(W) = I(V \setminus W)$$
 * 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 an algebraically closed field 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 $$(I : J^\infty )= \cup_{n \geq 1} (I:J^n)$$.

Examples

 * In $$\mathbb{Z}$$, $$((6):(2)) = (3)$$
 * In algebraic number theory, the ideal quotient is useful while studying fractional ideals. This is because the inverse of any invertible fractional ideal $$I$$ of an integral domain $$R$$ is given by the ideal quotient $$((1):I) = I^{-1}$$.
 * One geometric application of the ideal quotient is removing an irreducible component of an affine scheme. For example, let $$I = (xyz), J = (xy)$$ in $$\mathbb{C}[x,y,z]$$ be the ideals corresponding to the union of the x,y, and z-planes and x and y planes in $$\mathbb{A}^3_\mathbb{C}$$. Then, the ideal quotient $$(I:J) = (z)$$ is the ideal of the z-plane in $$\mathbb{A}^3_\mathbb{C}$$. 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 $$((x^4y^3):(x^2y^2)) = (x^2y)$$, 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 $$I \subset R[x_0,\ldots,x_n]$$ the saturation of $$I$$ is defined as the ideal quotient $$(I: \mathfrak{m}^\infty) = \cup_{i \geq 1} (I:\mathfrak{m}^i)$$ where $$\mathfrak{m} = (x_0,\ldots,x_n) \subset R[x_0,\ldots, x_n]$$. It is a theorem that the set of saturated ideals of $$R[x_0,\ldots, x_n]$$ contained in $$\mathfrak{m}$$ is in bijection with the set of projective subschemes in $$\mathbb{P}^n_R$$. This shows us that $$(x^4 + y^4 + z^4)\mathfrak{m}^k$$ defines the same projective curve as $$(x^4 + y^4 + z^4)$$ in $$\mathbb{P}^2_\mathbb{C}$$.