Editing Ideal class group (section)

== Definition ==

If <math>R</math> is an [[integral domain]], define a [[relation (mathematics)|relation]] <math>\sim</math> on nonzero [[fractional ideal]]s of <math>R</math> by <math>I\sim J</math> whenever there exist nonzero elements <math>a</math> and <math>b</math> of <math>R</math> such that <math>(a)I=(b)J</math>. It is easily shown that this is an [[equivalence relation]]. The equivalence classes are called the ''ideal classes'' of <math>R</math>.
Ideal classes can be multiplied: if <math>[I]</math> denotes the equivalence class of the ideal <math>I</math>, then the multiplication <math>[I][J]=[IJ]</math> is well-defined and [[commutative]]. The principal ideals form the ideal class <math>[R]</math> which serves as an [[identity element]] for this multiplication. Thus a class <math>[I]</math> has an [[inverse element|inverse]] <math>[J]</math> if and only if there is an ideal <math>J</math> such that <math>IJ</math> is a principal ideal. In general, such a <math>J</math> may not exist and consequently the set of ideal classes of <math>R</math> may only be a [[monoid]].

However, if <math>R</math> is the ring of [[algebraic integer]]s in an [[algebraic number field]], or more generally a [[Dedekind domain]], the multiplication defined above turns the set of fractional ideal classes into an [[abelian group]], the '''ideal class group''' of <math>R</math>. The group property of existence of inverse elements follows easily from the fact that, in a Dedekind domain, every non-zero ideal (except <math>R</math>) is a product of [[prime ideal]]s.