# 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.
• $J\subseteq I\Rightarrow (I:J)=R$
• $(I:R)=I$
• $(R:I)=R$
• $(I:(J+K))=(I:J)\cap (I:K)$
• $(I:(r))={\frac {1}{r}}(I\cap (r))$  (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, ..., 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)(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. 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 $(I:J^{\infty })=\cup _{n\geq 1}(I:J^{n})$ .

## Examples

• In $\mathbb {Z}$ , $((6):(2))=(3)$
• One geometric application of the ideal quotient is removing an irreducible component of an affine scheme. For example, let $I=(xyz),{\text{ }}J=(xy)$  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 $I\subset R[x_{0},\ldots ,x_{n}]$  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 $\mathbb {P} _{R}^{n}$ . 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 $\mathbb {P} _{\mathbb {C} }^{2}$ .