Ideal quotient

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

$(I : J) = \{r \in R | rJ \subset I\}$.

Then (I : J) is itself an ideal in R. The ideal quotient is viewed as a quotient because $IJ \subset K$ if and only if $I \subset K : J$. The ideal quotient is useful for calculating primary decompositions. It also arises in the description of the set difference in algebraic geometry.

Because of the notation, (I : J) is sometimes referred to as a colon ideal. There is an unrelated notion of the inverse of an ideal, known as a fractional ideal which is defined for Dedekind rings.

Properties

The ideal quotient satisfies the following properties:

• $J \subset I \Rightarrow I : J = R$
• I:R = I
• R:I = R
• $I : (J + K) = (I : J) \cap (I : K)$
• $I : (r) = \frac{1}{r}(I \cap (r))$ (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[x₁, ..., x_n], then

$I : J = (I : (g_1)) \cap (I : (g_2)) = (\frac{1}{g_1}(I \cap (g_1))) \cap (\frac{1}{g_2}(I \cap (g_2)))$

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

$I \cap (g_1) = tI + (1-t)(g_1) \cap k[x_1, \dots, x_n], \quad I \cap (g_2) = tI + (1-t)(g_1) \cap k[x_1, \dots, x_n]$

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 $I \cap (g_1)$.

Geometric interpretation

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, ..., x_n], then

Z(I : J) = cl(Z(I) \ Z(J)),

where "cl" denotes the Zariski closure, and Z denotes the taking of the variety defined by the ideal I.