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Then (I : J) is itself an ideal in R. The ideal quotient is viewed as a quotient because if and only if . 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:
- as -modules, where denotes the annihilator of as an -module.
- (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
Then elimination theory can be used to calculate the intersection of I with (g1) and (g2):
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 .
- 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], then
- Z(I : J) = cl(Z(I) \ Z(J))
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.