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.