|
We are creating the world's most trusted encyclopedia and knowledge base.
|
Dedekind domain
From Citizendium, the Citizens' Compendium
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
- 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.
- 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 K be a number field. Then the integral closure oKof
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.

