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

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

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

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

${\displaystyle I:J=(I:(g_{1}))\cap (I:(g_{2}))=\left({\frac {1}{g_{1}}}(I\cap (g_{1}))\right)\cap \left({\frac {1}{g_{2}}}(I\cap (g_{2}))\right)}$

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

${\displaystyle I\cap (g_{1})=tI+(1-t)(g_{1})\cap k,\quad I\cap (g_{2})=tI+(1-t)(g_{2})\cap k}$

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

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

${\displaystyle I(V):I(W)=I(V\setminus W)}$

where ${\displaystyle I(\bullet )}$ denotes the taking of the ideal associated to a subset.

• If I and J are ideals in k, with k an algebraically closed field and I radical then

${\displaystyle Z(I:J)=\mathrm {cl} (Z(I)\setminus Z(J))}$

where ${\displaystyle \mathrm {cl} (\bullet )}$ denotes the Zariski closure, and ${\displaystyle Z(\bullet )}$ 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:

${\displaystyle Z(I:J^{\infty })=\mathrm {cl} (Z(I)\setminus Z(J))}$

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

Examples

• In ${\displaystyle \mathbb {Z} }$ we have ${\displaystyle ((6):(2))=(3)}$.
• In algebraic number theory, the ideal quotient is useful while studying fractional ideals. This is because the inverse of any invertible fractional ideal ${\displaystyle I}$ of an integral domain ${\displaystyle R}$ is given by the ideal quotient ${\displaystyle ((1):I)=I^{-1}}$.
• One geometric application of the ideal quotient is removing an irreducible component of an affine scheme. For example, let ${\displaystyle I=(xyz),J=(xy)}$ in ${\displaystyle \mathbb {C} }$ be the ideals corresponding to the union of the x,y, and z-planes and x and y planes in ${\displaystyle \mathbb {A} _{\mathbb {C} }^{3}}$. Then, the ideal quotient ${\displaystyle (I:J)=(z)}$ is the ideal of the z-plane in ${\displaystyle \mathbb {A} _{\mathbb {C} }^{3}}$. 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 ${\displaystyle ((x^{4}y^{3}):(x^{2}y^{2}))=(x^{2}y)}$, 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 ${\displaystyle I\subset R}$ the saturation of ${\displaystyle I}$ is defined as the ideal quotient ${\displaystyle (I:{\mathfrak {m}}^{\infty })=\cup _{i\geq 1}(I:{\mathfrak {m}}^{i})}$ where ${\displaystyle {\mathfrak {m}}=(x_{0},\ldots ,x_{n})\subset R}$. It is a theorem that the set of saturated ideals of ${\displaystyle R}$ contained in ${\displaystyle {\mathfrak {m}}}$ is in bijection with the set of projective subschemes in ${\displaystyle \mathbb {P} _{R}^{n}}$. This shows us that ${\displaystyle (x^{4}+y^{4}+z^{4}){\mathfrak {m}}^{k}}$ defines the same projective curve as ${\displaystyle (x^{4}+y^{4}+z^{4})}$ in ${\displaystyle \mathbb {P} _{\mathbb {C} }^{2}}$.

References

1. David Cox; John Little; 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
2. Greuel, Gert-Martin; Pfister, Gerhard (2008). A Singular Introduction to Commutative Algebra (2nd ed.). Springer-Verlag. p. 485. ISBN 9783642442544.

• 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.

• Ideals (ring theory)