These axioms are sufficient for ensuring that every ideal of that is not or can be written as a finite product of prime ideals in a unique way (up to a permutation of the terms of the product). In fact, this property has a natural extension to the fractional ideals of the field of fractions of .
This product extends to the set of fractional ideals of the field (i.e., the nonzero finitely generated -submodules of ).
- Every principal ideal domain is a unique factorization domain, but the converse is not true in general. However, these notions are equivalent for Dedekind domains; that is, a Dedekind domain is a principal ideal domain if and only if it is a unique factorization domain.
- The localization of a Dedekind domain at a non-zero prime ideal is a principal ideal domain, which is either a field or a discrete valuation ring.
- The ring is a Dedekind domain.
- Let be an algebraic number field. Then the integral closure of in is again a Dedekind domain. In fact, if is a Dedekind domain with field of fractions , and is a finite extension of and is the integral closure of in , then is again a Dedekind domain.