Ideal quotient

Not to be confused with the quotient of a ring by an ideal.

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

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

Then ${\displaystyle (I:J)}$ is itself an ideal in ${\displaystyle R}$. 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).

${\displaystyle (I:J)}$ 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 ${\displaystyle R}$ 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 ${\displaystyle I=(f_{1},f_{2},f_{3})}$ and ${\displaystyle J=(g_{1},g_{2})}$ are ideals in ${\displaystyle \mathbb {k} [x_{1},\ldots ,x_{n}]}$, 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 ${\displaystyle I}$ with ${\displaystyle (g_{1})}$ and ${\displaystyle (g_{2})}$:

${\displaystyle I\cap (g_{1})=tI+(1-t)(g_{1})\cap \mathbb {k} [x_{1},\dots ,x_{n}],\quad I\cap (g_{2})=tI+(1-t)(g_{2})\cap \mathbb {k} [x_{1},\dots ,x_{n}]}$

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. ^[1] More precisely,

• If ${\displaystyle W}$ is an affine variety (not necessarily irreducible) and ${\displaystyle V}$ 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 ${\displaystyle I}$ and ${\displaystyle J}$ are ideals in ${\displaystyle \mathbb {k} [x_{1},\ldots ,x_{n}]}$, with ${\displaystyle \mathbb {k} }$ an algebraically closed field and ${\displaystyle I}$ 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 ${\displaystyle I}$ is not radical, then the same property holds if we saturate the ideal ${\displaystyle J}$:

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

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 ${\displaystyle \mathbb {C} [x,y,z]}$ 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 ${\displaystyle I\subset R[x_{0},\ldots ,x_{n}]}$ 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 ${\displaystyle {\mathfrak {m}}=(x_{0},\ldots ,x_{n})\subset R[x_{0},\ldots ,x_{n}]}$. It is a theorem that the set of saturated ideals of ${\displaystyle R[x_{0},\ldots ,x_{n}]}$ contained in ${\displaystyle {\mathfrak {m}}}$ is in bijection with the set of projective subschemes in ${\displaystyle \mathbb {P} _{R}^{n}}$. ^[2] 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 ${\displaystyle \mathbb {P} _{\mathbb {C} }^{2}}$.

Notes

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.

References

• Ene, Viviana; Herzog, Jürgen (2011). Gröbner Bases in Commutative Algebra. Graduate Studies in Mathematics. Vol. 130. American Mathematical Society. ISBN   0-8218-7287-7.
• Atiyah, M.F.; Macdonald, I.G. (1994). Introduction to Commutative Algebra. Addison-Wesley. ISBN   0-201-40751-5.