if "r" is an integral domain, define a relation ~ on nonzero fractional ideals of "r" by "i" ~ "j" whenever there exist nonzero elements "a" and "b" of "r" such that ("a")"i" = ("b")"j". (here the notation ("a") means the principal ideal of "r" consisting of all the multiples of "a".) it is easily shown that this is an equivalence relation. the equivalence classes are called the "ideal classes" of "r".