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) = \$

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:
*$(I :J)=\mathrm_R((J+I)/I)$ as $R$-modules, where $\mathrm_R(M)$ denotes the annihilator of $M$ as an $R$-module.
*$J \subseteq I \Leftrightarrow (I : J) = R$ (in particular, $(I : I) = (R : I) = (I : 0) = R$)
*$(I : R) = I$
*$(I : (JK)) = ((I : J) : K)$
*$(I : (J + K)) = (I : J) \cap (I : K)$
*$((I \cap J) : K) = (I : K) \cap (J : K)$
*$(I : (r)) = \frac(I \cap (r))$ (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'' = (''f''₁, ''f''₂, ''f''₃) and ''J'' = (''g''₁, ''g''₂) are ideals in ''k'' 'x''₁, ..., ''x''_''n'' then
:$I : J = (I : (g_1)) \cap (I : (g_2)) = \left(\frac(I \cap (g_1))\right) \cap \left(\frac(I \cap (g_2))\right)$

Then elimination theory can be used to calculate the intersection of ''I'' with (''g''₁) and (''g''₂):
:I \cap (g_1) = tI + (1-t) (g_1) \cap k _1, \dots, x_n \quad I \cap (g_2) = tI + (1-t) (g_2) \cap k _1, \dots, x_n/math>

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. 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
::$Z(I : J) = \mathrm(Z(I) \setminus Z(J))$

:where $\mathrm(\bullet)$ 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'':
::$Z(I : J^) = \mathrm(Z(I) \setminus Z(J))$
:where $(I : J^\infty )= \cup_ (I:J^n)$.

Examples

* In $\mathbb$, $((6):(2)) = (3)$
* 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 $((1):I) = I^$.
* One geometric application of the ideal quotient is removing an irreducible component of an affine scheme. For example, let $I = (xyz), J = (xy)$ in \mathbb ,y,z/math> be the ideals corresponding to the union of the x,y, and z-planes and x and y planes in $\mathbb^3_\mathbb$. Then, the ideal quotient $(I:J) = (z)$ is the ideal of the z-plane in $\mathbb^3_\mathbb$. 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 _0,\ldots,x_n/math> the saturation of $I$ is defined as the ideal quotient $(I: \mathfrak^\infty) = \cup_ (I:\mathfrak^i)$ where \mathfrak = (x_0,\ldots,x_n) \subset R _0,\ldots, x_n/math>. It is a theorem that the set of saturated ideals of R _0,\ldots, x_n/math> contained in $\mathfrak$ is in bijection with the set of projective subschemes in $\mathbb^n_R$. This shows us that $(x^4 + y^4 + z^4)\mathfrak^k$ defines the same projective curve as $(x^4 + y^4 + z^4)$ in $\mathbb^2_\mathbb$.

References

{{Reflist
* 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.

Ideals (ring theory)