Editing Ideal class group (section)

==Relation with the group of units==
It was remarked above that the ideal class group provides part of the answer to the question of how much ideals in a [[Dedekind domain]] behave like elements. The other part of the answer is provided by the [[group of units]] of the Dedekind domain, since passage from principal ideals to their generators requires the use of [[unit (ring theory)|units]] (and this is the rest of the reason for introducing the concept of fractional ideal, as well).

Define a map from <math>R^\times</math> to the set of all nonzero fractional ideals of <math>R</math> by sending every element to the principal (fractional) ideal it generates. This is a [[group homomorphism]]; its [[Kernel (algebra)|kernel]] is the group of units of <math>R</math>, and its [[cokernel]] is the ideal class group of <math>R</math>. The failure of these groups to be trivial is a measure of the failure of the map to be an isomorphism: that is the failure of ideals to act like ring elements, that is to say, like numbers.