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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).
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(''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''<sub>1</sub>, ''f''<sub>2</sub>, ''f''<sub>3</sub>) and ''J'' = (''g''<sub>1</sub>, ''g''<sub>2</sub>) are ideals in ''k''[''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>], 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''<sub>1</sub>) and (''g''<sub>2</sub>):
:<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''<sub>1</sub>) 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''<sub>1</sub>, ..., ''x''<sub>''n''</sub>], 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]]

Latest revision as of 21:18, 2 July 2014


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