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 17:04, 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.
, and let 
be any nonzero ideal of 
. We call an ideal 
. (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 There are also various properties of homological importance that Dedekind domains satisfy.
