Ideal quotient

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

[math]\displaystyle{ (I : J) = \{r \in R \mid rJ \subseteq I\} }[/math]

Then (I : J) is itself an ideal in R. The ideal quotient is viewed as a quotient because [math]\displaystyle{ KJ \subseteq I }[/math] if and only if [math]\displaystyle{ K \subseteq (I : J) }[/math]. 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:

• [math]\displaystyle{ (I :J)=\mathrm{Ann}_R((J+I)/I) }[/math] as [math]\displaystyle{ R }[/math]-modules, where [math]\displaystyle{ \mathrm{Ann}_R(M) }[/math] denotes the annihilator of [math]\displaystyle{ M }[/math] as an [math]\displaystyle{ R }[/math]-module.
• [math]\displaystyle{ J \subseteq I \Leftrightarrow (I : J) = R }[/math] (in particular, [math]\displaystyle{ (I : I) = (R : I) = (I : 0) = R }[/math])
• [math]\displaystyle{ (I : R) = I }[/math]
• [math]\displaystyle{ (I : (JK)) = ((I : J) : K) }[/math]
• [math]\displaystyle{ (I : (J + K)) = (I : J) \cap (I : K) }[/math]
• [math]\displaystyle{ ((I \cap J) : K) = (I : K) \cap (J : K) }[/math]
• [math]\displaystyle{ (I : (r)) = \frac{1}{r}(I \cap (r)) }[/math] (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

[math]\displaystyle{ 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) }[/math]

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

[math]\displaystyle{ 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] }[/math]

Calculate a Gröbner basis for [math]\displaystyle{ tI+(1-t)(g_1) }[/math] with respect to lexicographic order. Then the basis functions which have no t in them generate [math]\displaystyle{ I \cap (g_1) }[/math].

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

[math]\displaystyle{ I(V) : I(W) = I(V \setminus W) }[/math]

where [math]\displaystyle{ I(\bullet) }[/math] denotes the taking of the ideal associated to a subset.

• If I and J are ideals in k[x₁, ..., x_n], with k an algebraically closed field and I radical then

[math]\displaystyle{ Z(I : J) = \mathrm{cl}(Z(I) \setminus Z(J)) }[/math]

where [math]\displaystyle{ \mathrm{cl}(\bullet) }[/math] denotes the Zariski closure, and [math]\displaystyle{ Z(\bullet) }[/math] 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:

[math]\displaystyle{ Z(I : J^{\infty}) = \mathrm{cl}(Z(I) \setminus Z(J)) }[/math]

where [math]\displaystyle{ (I : J^\infty )= \cup_{n \geq 1} (I:J^n) }[/math].

Examples

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

References

• 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.