Dedekind ideal
E634837
A Dedekind ideal is a type of ideal in ring theory central to algebraic number theory, particularly in the study of Dedekind domains and unique factorization of ideals.
Statements (42)
| Predicate | Object |
|---|---|
| instanceOf |
ideal in ring theory
ⓘ
mathematical concept ⓘ |
| appearsIn |
algebraic number theory textbooks
ⓘ
commutative algebra textbooks ⓘ |
| field |
algebraic number theory
ⓘ
ring theory ⓘ |
| hasProperty |
every nonzero proper ideal factors uniquely into prime ideals in a Dedekind domain
ⓘ
nonzero ideals correspond to finitely generated torsion modules over a Dedekind domain ⓘ nonzero ideals in a Dedekind domain are invertible fractional ideals ⓘ nonzero ideals in a Dedekind domain have unique factorization up to order ⓘ nonzero prime ideals in a Dedekind domain are maximal ⓘ |
| isDefinedIn | commutative ring with identity ⓘ |
| isNamedAfter | Richard Dedekind NERFINISHED ⓘ |
| isRelatedTo |
Dedekind domain
NERFINISHED
ⓘ
Krull dimension one ⓘ Noetherian ring NERFINISHED ⓘ class number ⓘ fractional ideal ⓘ fractional ideal group ⓘ ideal class group ⓘ ideal factorization ⓘ ideal inversion ⓘ ideal multiplication ⓘ ideal norm ⓘ integral ideal ⓘ integrally closed domain ⓘ localization of rings ⓘ maximal ideal ⓘ prime decomposition in number fields ⓘ prime ideal ⓘ principal ideal ⓘ principal ideal domain ⓘ unique factorization domain ⓘ unique factorization of ideals ⓘ valuation theory ⓘ |
| isStudiedIn | Dedekind domain NERFINISHED ⓘ |
| isUsedFor |
decomposing prime ideals in extensions of number fields
ⓘ
defining ideal class group ⓘ studying arithmetic of algebraic number fields ⓘ studying failure of unique factorization of elements ⓘ studying ramification in number fields ⓘ studying splitting of primes in extensions ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.