Ideal quotient

The ideal quotient satisfies the following properties:

• $(I\; :J)=\backslash mathrm\{Ann\}\_R((J+I)/I)$ as $R$-modules, where $\backslash mathrm\{Ann\}\_R(M)$ denotes the annihilator of $M$ as an $R$-module.
• $J\; \backslash subseteq\; I\; \backslash Leftrightarrow\; (I\; :\; J)\; =\; R$ (in particular, $(I\; :\; I)\; =\; (R\; :\; I)\; =\; (I\; :\; 0)\; =\; R$)
• $(I\; :\; R)\; =\; I$
• $(I\; :\; (JK))\; =\; ((I\; :\; J)\; :\; K)$
• $(I\; :\; (J\; +\; K))\; =\; (I\; :\; J)\; \backslash cap\; (I\; :\; K)$
• $((I\; \backslash cap\; J)\; :\; K)\; =\; (I\; :\; K)\; \backslash cap\; (J\; :\; K)$
• $(I\; :\; (r))\; =\; \backslash frac\{1\}\{r\}(I\; \backslash cap\; (r))$ (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, ..., x_n], then

:$I\; :\; J\; =\; (I\; :\; (g\_1))\; \backslash cap\; (I\; :\; (g\_2))\; =\; \backslash left(\backslash frac\{1\}\{g\_1\}(I\; \backslash cap\; (g\_1))\backslash right)\; \backslash cap\; \backslash left(\backslash frac\{1\}\{g\_2\}(I\; \backslash cap\; (g\_2))\backslash right)$

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

:$I\; \backslash cap\; (g\_1)\; =\; tI\; +\; (1-t)\; (g\_1)\; \backslash cap\; k[x\_1,\; \backslash dots,\; x\_n],\; \backslash quad\; I\; \backslash cap\; (g\_2)\; =\; tI\; +\; (1-t)\; (g\_2)\; \backslash cap\; k[x\_1,\; \backslash dots,\; x\_n]$

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 $I\; \backslash cap\; (g\_1)$.

The ideal quotient corresponds to set difference in algebraic geometry.{{cite book |author1=David Cox |author2=John Little |author3=Donal O'Shea |year=1997 |title=Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra |publisher=Springer |isbn=0-387-94680-2}}, p.195 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

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

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

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

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

:where $\backslash mathrm\{cl\}(\backslash bullet)$ denotes the Zariski closure, and $Z(\backslash bullet)$ 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:

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

:where $(I\; :\; J^\backslash infty\; )=\; \backslash cup\_\{n\; \backslash geq\; 1\}\; (I:J^n)$.

• In $\backslash mathbb\{Z\}$ we have $((6):(2))\; =\; (3)$.
• 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 $((1):I)\; =\; I^\{-1\}$.
• One geometric application of the ideal quotient is removing an irreducible component of an affine scheme. For example, let $I\; =\; (xyz),\; J\; =\; (xy)$ in $\backslash mathbb\{C\}[x,y,z]$ be the ideals corresponding to the union of the x,y, and z-planes and x and y planes in $\backslash mathbb\{A\}^3\_\backslash mathbb\{C\}$. Then, the ideal quotient $(I:J)\; =\; (z)$ is the ideal of the z-plane in $\backslash mathbb\{A\}^3\_\backslash mathbb\{C\}$. 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 $I\; \backslash subset\; R[x\_0,\backslash ldots,x\_n]$ the saturation of $I$ is defined as the ideal quotient $(I:\; \backslash mathfrak\{m\}^\backslash infty)\; =\; \backslash cup\_\{i\; \backslash geq\; 1\}\; (I:\backslash mathfrak\{m\}^i)$ where $\backslash mathfrak\{m\}\; =\; (x\_0,\backslash ldots,x\_n)\; \backslash subset\; R[x\_0,\backslash ldots,\; x\_n]$. It is a theorem that the set of saturated ideals of $R[x\_0,\backslash ldots,\; x\_n]$ contained in $\backslash mathfrak\{m\}$ is in bijection with the set of projective subschemes in $\backslash mathbb\{P\}^n\_R$.{{Cite book|title=A Singular Introduction to Commutative Algebra|url=https://archive.org/details/singularintroduc00greu_498|url-access=limited|last=Greuel|first=Gert-Martin|last2=Pfister|first2=Gerhard|publisher=Springer-Verlag|year=2008|isbn=9783642442544|edition=2nd|page=[https://archive.org/details/singularintroduc00greu_498/page/n500 485]}} This shows us that $(x^4\; +\; y^4\; +\; z^4)\backslash mathfrak\{m\}^k$ defines the same projective curve as $(x^4\; +\; y^4\; +\; z^4)$ in $\backslash mathbb\{P\}^2\_\backslash mathbb\{C\}$.

{{Reflist}}

• {{Cite book |first1=Viviana |last1=Ene |first2=Jürgen |last2=Herzog |title=Gröbner Bases in Commutative Algebra |publisher=American Mathematical Society |series=Graduate Studies in Mathematics |volume=130 |year=2011 |isbn=0-8218-7287-7 }}
• {{Cite book|last1=Atiyah |first1=M.F. |title=Introduction to Commutative Algebra |title-link=Introduction to Commutative Algebra |last2=Macdonald |first2=I.G. |publisher=Addison-Wesley |year=1994 |isbn=0-201-40751-5 |author-link=Michael Atiyah |author-link2=Ian G. Macdonald}}

Category:Ideals (ring theory)