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.