Ideal quotient explained

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) = Ann _{R((J+I)/I)} as R -modules, where Ann _{R(M)} denotes the annihilator of M as an R -module.J \subseteq I \Leftrightarrow (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)) = (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[''x''₁, ..., ''x''_''n''], then

I : J = (I : (g _{1))} \cap (I : (g _{2))} = \left( (1)/(g _{1}) (I \cap (g _{1))\right)} \cap \left( (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,} ... , x _{n],} I \cap (g _{2)} = tI + ( 1- t) (g _{2)} \cap k[x _{1,} ... , 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.^[1] 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[''x''₁, ..., ''x''_''n''], with k an algebraically closed field and I radical then

Z(I : J) = cl (Z(I) \setminus Z(J))

where

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} ) = cl (Z(I) \setminus Z(J))

where

(I : J ^{infty} ) = \cup _{n} (I:J ^{n)} .

Examples

• In

Z we have (( 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 C [x,y,z] be the ideals corresponding to the union of the x,y, and z-planes and x and y planes in A ^{3} _{C} . Then, the ideal quotient (I:J) = (z) is the ideal of the z-plane in A ^{3} _{C} . 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 ^{4y} ^{3):(x} ^{2y} ^{2))} = (x ^{2y)} , 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: ak{m} ^{infty)} = \cup _{i} (I: ak{m} ^{i)} where ak{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 ak{m} is in bijection with the set of projective subschemes in P ^{n} _{R} .^[2] This shows us that (x ^{4} + y ^{4} + z ^{4)ak{m}} ^{k} defines the same projective curve as (x ^{4} + y ^{4} + z ^{4)} in P ^{2} _{C} .

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.

Notes and References

1. Book: David Cox . John Little . Donal O'Shea . 1997 . Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra . Springer . 0-387-94680-2., p.195
2. Book: Greuel, Gert-Martin. A Singular Introduction to Commutative Algebra. limited. Pfister. Gerhard. Springer-Verlag. 2008. 9783642442544. 2nd. 485.