Define a map from "R"× to the set of all nonzero fractional ideals of "R" by sending every element to the principal (fractional) ideal it generates. This is a group homomorphism; its kernel is the group of units of "R", and its cokernel is the ideal class group of "R". 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.