Difference between revisions of "Dedekind domain"

m (I think an integral domain is usually defined to be commutative)
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A '''Dedekind domain''' is a [[commutative]] [[integral domain]] <math>R</math> satisfying the following properties:
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 $R$ satisfying the following properties:

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 $R$ be a Dedekind domain with field of fractions $K$, and let $I$ be any nonzero ideal of $R$. Then set $I^{-1}=\{a\in K\mid aI\subseteq R\}$. We call an ideal $I$ invertible if $II^{-1}=R$. (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 $R$ is a Dedekind domain. This is sometimes used as a definition.

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

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