# Dedekind domain

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Revision as of 05:33, 24 June 2011 by Boris Tsirelson (talk | contributions)

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* = *F**r**a**c*(*o*) (i.e., the nonzero finitely generated *o*-submodules of *K*).

## [edit] 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.

## [edit] Examples

- The ring is a Dedekind domain.
- Let
*K*be an algebraic number field. Then the integral closure*o*_{K}of 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.

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