somewhat later again dedekind formulated the concept of ideal, kummer having worked in a different way. at this point the existing examples could be unified. it was shown that while rings of algebraic integers do not always have unique factorization into primes (because they need not be principal ideal domains), they do have the property that every proper ideal admits a unique factorization as a product of prime ideals (that is, every ring of algebraic integers is a dedekind domain). the size of the ideal class group can be considered as a measure for the deviation of a ring from being a principal ideal domain; a ring is a principal domain if and only if it has a trivial ideal class group.