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:

• $(I:J)={\mathrm{A}\mathrm{n}\mathrm{n}}_R((J+I)/I)$ as $R$-modules, where ${\mathrm{A}\mathrm{n}\mathrm{n}}_R(M)$ denotes the annihilator of $M$ as an $R$-module.
• $J\subseteq I\iff (I:J)=R$ (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, ..., x_n], then

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

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

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

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, ..., x_n], with k an algebraically closed field and I radical then

$Z(I:J)=\mathrm{c}\mathrm{l}(Z(I)\setminus Z(J))$

where $\mathrm{c}\mathrm{l}(\bullet )$ 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:

$Z(I:J^{\mathrm{\infty }})=\mathrm{c}\mathrm{l}(Z(I)\setminus Z(J))$

where $(I:J^{\mathrm{\infty }})={\cup }_{n\ge 1}(I:J^n)$.

Examples

• In $\mathbb{\mathbb{Z}}$, $((6):(2))=(3)$
• 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 ${\mathbb{𝔸}}_{\mathbb{ℂ}}^3$. Then, the ideal quotient $(I:J)=(z)$ is the ideal of the z-plane in ${\mathbb{𝔸}}_{\mathbb{ℂ}}^3$. 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 $((x^4{y}^3):(x^2{y}^2))=(x^{2}y)$, 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 $I\subset R[x_0,\dots ,x_n]$ the saturation of $I$ is defined as the ideal quotient $(I:{\mathfrak{𝔪}}^{\mathrm{\infty }})={\cup }_{i\ge 1}(I:{\mathfrak{𝔪}}^i)$ where $\mathfrak{𝔪}=(x_0,\dots ,x_n)\subset R[x_0,\dots ,x_n]$. It is a theorem that the set of saturated ideals of $R[x_0,\dots ,x_n]$ contained in $\mathfrak{𝔪}$ is in bijection with the set of projective subschemes in ${\mathbb{\mathbb{P}}}_R^n$. 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 ${\mathbb{\mathbb{P}}}_{\mathbb{ℂ}}^2$.

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.