Principal fractional ideal ($a{\mathcal{O}}_K$)

A principal fractional ideal of a number field is $a{\mathcal{O}}_K$ for a nonzero $a\in K$. Its generators differ by a unit in the ring of integers of a number field. Quotienting nonzero fractional ideals by principal fractional ideals gives the ideal class group; requiring congruences and signs of the generators instead gives a ray class group.