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 ∈ R ∣ r J ⊂ I } Then (I : J) is itself an ideal in R. The ideal quotient is viewed as a quotient because I J ⊂ K if and only if I ⊂ K : 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.
The ideal quotient satisfies the following properties:
( I : J ) = A n n R ( ( J + I ) / I ) as R -modules, where A n n R ( M ) denotes the annihilator of M as an R -module. J ⊂ I ⇒ I : J = R I : R = I R : I = R I : ( J + K ) = ( I : J ) ∩ ( I : K ) I : ( r ) = 1 r ( I ∩ ( r ) ) (as long as R is an integral domain)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 ) ) ∩ ( I : ( g 2 ) ) = ( 1 g 1 ( I ∩ ( g 1 ) ) ) ∩ ( 1 g 2 ( I ∩ ( g 2 ) ) ) Then elimination theory can be used to calculate the intersection of I with (g1) and (g2):
I ∩ ( g 1 ) = t I + ( 1 − t ) ( g 1 ) ∩ k [ x 1 , … , x n ] , I ∩ ( g 2 ) = t I + ( 1 − t ) ( g 1 ) ∩ k [ x 1 , … , 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 ∩ ( g 1 ) .
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 ∖ W ) where I ( ∙ ) 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 ) = c l ( Z ( I ) ∖ Z ( J ) ) where c l ( ∙ ) denotes the Zariski closure, and Z ( ∙ ) 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 ∞ ) = c l ( Z ( I ) ∖ Z ( J ) ) where ( I : J ∞ ) = ∪ n ≥ 1 ( I : J n ) .
One geometric application of the ideal quotient is removing an irreducible component of an affine scheme. For example, let I = ( x y z ) , J = ( x y ) 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 C 3 . Then, the ideal quotient ( I : J ) = ( z ) is the ideal of the z-plane in A 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 ⊂ R [ x 0 , … , x n ] the saturation of I is defined as the ideal quotient ( I : m ∞ ) = ∪ i ≥ 1 ( I : m i ) where m = ( x 0 , … , x n ) ⊂ R [ x 0 , … , x n ] . It is a theorem that the set of saturated ideals of R [ x 0 , … , x n ] contained in m is in bijection with the set of projective subschemes in P R n . This shows us that ( x 4 + y 4 + z 4 ) m k defines the same projective curve as ( x 4 + y 4 + z 4 ) in P C 2 .