Dedekind domain: Difference between revisions
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A '''Dedekind domain''' is a [[Noetherian domain]] <math>o</math>, integrally closed in its [[field of fractions]], so that every [[prime ideal]] is maximal. | |||
These axioms are sufficient for ensuring that every [[ideal (mathematics)|ideal]] of <math>o</math> that is not <math>(0)</math> or <math>(1)</math> can be written as a finite product of prime ideals in a unique way (up to a permutation of the terms of the product). In fact, this property has a natural extension to the fractional ideals of the field of fractions of <math>o</math>. | |||
These axioms are sufficient for ensuring that every ideal of <math>o</math> that is not <math>(0)</math> or <math>(1)</math> can be written as a finite product of prime ideals in a unique way (up to a permutation of the terms of the product). In fact, this property has a natural extension to the fractional ideals of the field of fractions of <math>o</math>. | |||
This product extends to the set of fractional ideals of the field <math>K=Frac(o)</math> (i.e., the nonzero finitely generated <math>o</math>-submodules of <math>K</math>). | This product extends to the set of fractional ideals of the field <math>K=Frac(o)</math> (i.e., the nonzero finitely generated <math>o</math>-submodules of <math>K</math>). | ||
==Useful | ==Useful properties== | ||
#Every principal ideal domain is a unique factorization domain, but the converse is not true in general. However, these notions are equivalent for Dedekind domains. | #Every [[principal ideal domain]] is a [[unique factorization domain]], but the converse is not true in general. However, these notions are equivalent for Dedekind domains; that is, a Dedekind domain <math>A</math> is a principal ideal domain if and only if it is a unique factorization domain. | ||
#The localization of a Dedekind domain at a non-zero prime ideal is a principal ideal domain, which is either a field or a discrete valuation ring. | #The [[localization]] of a Dedekind domain at a non-zero prime ideal is a principal ideal domain, which is either a [[field (mathematics)|field]] or a [[discrete valuation ring]]. | ||
==Examples== | ==Examples== | ||
#The ring <math>\mathbb{Z}</math> is a Dedekind domain. | #The ring <math>\mathbb{Z}</math> is a Dedekind domain. | ||
#Let <math>K</math> be | #Let <math>K</math> be an [[algebraic number field]]. Then the integral closure <math>o_K</math>of <math>\mathbb{Z}</math> in <math>K</math> is again a Dedekind domain. In fact, if <math>o</math> is a Dedekind domain with field of fractions <math>K</math>, and <math>L/K</math> is a finite extension of <math>K</math> and <math>O</math> is the integral closure of <math>o</math> in <math>L</math>, then <math>O</math> is again a Dedekind domain. | ||
[[Category: | ==Finitely generated modules over a Dedekind domain==[[Category:Suggestion Bot Tag]] | ||
Latest revision as of 11:01, 5 August 2024
A Dedekind domain is a Noetherian domain Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle o} , integrally closed in its field of fractions, so that every prime ideal is maximal.
These axioms are sufficient for ensuring that every ideal of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle o} that is not or can be written as a finite product of prime ideals in a unique way (up to a permutation of the terms of the product). In fact, this property has a natural extension to the fractional ideals of the field of fractions of .
This product extends to the set of fractional ideals of the field (i.e., the nonzero finitely generated -submodules of ).
Useful properties
- Every principal ideal domain is a unique factorization domain, but the converse is not true in general. However, these notions are equivalent for Dedekind domains; that is, a Dedekind domain is a principal ideal domain if and only if it is a unique factorization domain.
- The localization of a Dedekind domain at a non-zero prime ideal is a principal ideal domain, which is either a field or a discrete valuation ring.
Examples
- The ring is a Dedekind domain.
- Let be an algebraic number field. Then the integral closure of in is again a Dedekind domain. In fact, if is a Dedekind domain with field of fractions , and is a finite extension of and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle O} is the integral closure of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle o} in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle O} is again a Dedekind domain.
==Finitely generated modules over a Dedekind domain==