Dedekind domain

A Dedekind domain is a Noetherian domain $$o$$, integrally closed in its field of fractions, so that every prime ideal is maximal.

These axioms are sufficient for ensuring that every ideal of $$o$$ that is not $$(0)$$ or $$(1)$$ 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 $$o$$.

This product extends to the set of fractional ideals of the field $$K=Frac(o)$$ (i.e., the nonzero finitely generated $$o$$-submodules of $$K$$).

Useful properties

 * 1) 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 $$A$$ is a principal ideal domain if and only if it is a unique factorization domain.
 * 2) 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.

Examples

 * 1) The ring $$\mathbb{Z}$$ is a Dedekind domain.
 * 2) Let $$K$$ be an algebraic number field. Then the integral closure $$o_K$$of $$\mathbb{Z}$$ in $$K$$ is again a Dedekind domain. In fact, if $$o$$ is a Dedekind domain with field of fractions $$K$$, and $$L/K$$ is a finite extension of $$K$$ and $$O$$ is the integral closure of $$o$$ in $$L$$, then $$O$$ is again a Dedekind domain.