Dedekind domain
From Citizendium, the Citizens' Compendium
A Dedekind domain is a Noetherian domain
, integrally closed in its field of fractions, so that every prime ideal is maximal.
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
).
Useful properties
- 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.
Examples
- 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.

