Difference between revisions of "Dedekind domain"

Revision as of 13:33, 10 December 2007

A Dedekind domain is a commutative 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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 * Dedekind domains have unique prime factorizations of [[ideal]]s (but not necessarily of elements).
-* Ideals are invertible. Let <math>R</math> be a Dedekind domain with field of fractions <math>K</math>, and let <math>I</math> be any nonzero ideal of <math>R</math>. Then set <math>I^{-1}=\{a\in K\mid aI\subseteq R\}</math>. We call an ideal <math>I</math> '''invertible''' if <math>II^{-1}=R</math>. (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 <math>R</math> is a Dedekind domain. This is sometimes used as a definition.
+* Ideals are invertible if we extend to [[fractional ideal]]s. Let <math>R</math> be a Dedekind domain with field of fractions <math>K</math>, and let <math>I</math> be any nonzero ideal of <math>R</math>. Then set <math>I^{-1}=\{a\in K\mid aI\subseteq R\}</math>. We call an ideal <math>I</math> '''invertible''' if <math>II^{-1}=R</math>. (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 <math>R</math> is a Dedekind domain. This is sometimes used as a definition.
 There are also various properties of [[homological algebra|homological]] importance that Dedekind domains satisfy.