Graham number

2013-12-23T11:53:36Z

92.19.239.125:

{{Unreferenced| date=August 2012}}
In [[abstract algebra]], if ''I'' and ''J'' are [[ideal (ring theory)|ideals]] of a commutative [[ring (mathematics)|ring]] ''R'', their '''ideal quotient''' (''I'' : ''J'') is the set

:<math>(I : J) = \{r \in R | rJ \subset I\}</math>

Then (''I'' : ''J'') is itself an ideal in ''R''. The ideal quotient is viewed as a quotient because <math>IJ \subset K</math> if and only if <math>I \subset K : J</math>. The ideal quotient is useful for calculating [[primary decomposition]]s. It also arises in the description of the [[Complement (set theory)#Relative complement|set difference]] in [[algebraic geometry]] (see below).

(''I'' : ''J'') is sometimes referred to as a '''colon ideal''' because of the notation. 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:
*<math>(I :J)=\mathrm{Ann}_R((J+I)/I)</math> as <math>R</math>-modules, where <math>\mathrm{Ann}_R(M)</math> denotes the [[annihilator (ring theory)|annihilator]] of <math>M</math> as an <math>R</math>-module.
*<math>J \subset I \Rightarrow I : J = R</math>
*<math>I : R = I</math>
*<math>R : I = R</math>
*<math>I : (J + K) = (I : J) \cap (I : K)</math>
*<math>I : (r) = \frac{1}{r}(I \cap (r))</math> (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
:<math>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)</math>

Then elimination theory can be used to calculate the intersection of ''I'' with (''g''1) and (''g''2):
:<math>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]</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 <math>I \cap (g_1)</math>.

==Geometric interpretation==
The ideal quotient corresponds to [[Complement (set theory)#Relative complement|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''[''x''1, ..., ''x''''n''], then
:''Z''(''I'' : ''J'') = cl(''Z''(''I'') \ ''Z''(''J''))

where "cl" denotes the [[Zariski topology|Zariski]] [[Closure (topology)|closure]], and ''Z'' denotes the taking of the variety defined by the ideal ''I''.

==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.
[[Category:Ideals]]

formulasearchengine - User contributions [en]

Graham number