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 (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:

• $(I:J)=\mathrm {Ann} _{R}((J+I)/I)$ as $R$-modules, where $\mathrm {Ann} _{R}(M)$ denotes the annihilator of $M$ as an $R$-module.
• $J\subset I\Rightarrow I:J=R$
• $I:R=I$
• $R:I=R$
• $I:(J+K)=(I:J)\cap (I:K)$
• $I:(r)={\frac {1}{r}}(I\cap (r))$ (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

$I:J=(I:(g_{1}))\cap (I:(g_{2}))=\left({\frac {1}{g_{1}}}(I\cap (g_{1}))\right)\cap \left({\frac {1}{g_{2}}}(I\cap (g_{2}))\right)$

Then elimination theory can be used to calculate the intersection of 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

$I(V):I(W)=I(V\setminus W)$

where $I(\bullet )$ denotes the taking of the ideal associated to a subset.

• If I and J are ideals in k[x₁, ..., x_n], with k algebraically closed and I radical then

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

where $\mathrm {cl} (\bullet )$ 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:

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

where $J^{\infty }=J+J^{2}+\cdots +J^{n}+\cdots$.

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.