# Difference between revisions of "Dedekind domain"

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− | A '''Dedekind domain''' is a | + | A '''Dedekind domain''' is a [[integral domain]] <math>R</math> satisfying the following properties: |

* <math>R</math> is a [[noetherian]] [[ring]]. | * <math>R</math> is a [[noetherian]] [[ring]]. |

## Revision as of 16:04, 5 September 2008

A **Dedekind domain** is a integral domain satisfying the following properties:

- is a noetherian ring.
- Every prime ideal of is a maximal ideal.
- is integrally closed in its field of fractions.

Dedekind domains are very important in abstract algebra and number theory. For example, the ring of integers of any number field is a Dedekind domain.

There are several very nice properties of Dedekind domains:

- Dedekind domains have unique prime factorizations of ideals (but not necessarily of elements).
- Ideals are invertible if we extend to fractional ideals. Let be a Dedekind domain with field of fractions , and let be any nonzero ideal of . Then set . We call an ideal
**invertible**if . (Note that this is always a subset, but it is not always equal unless we are in a Dedekind domain.) In fact, the converse is true as well: if all nonzero ideals are invertible, then is a Dedekind domain. This is sometimes used as a definition.

There are also various properties of homological importance that Dedekind domains satisfy.