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 a 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.

References

• Neukirch, Jürgen (1999). Algebraic Number Theory, , ISBN 3-540-65399-6.
• Neukirch, Jürgen (1999). Algebraic Number Theory. Springer-Verlag. ISBN 3-540-65399-6.