Statements (28)
| Predicate | Object |
|---|---|
| gptkbp:instance_of |
gptkb:item
|
| gptkbp:bfsLayer |
6
|
| gptkbp:bfsParent |
gptkb:Dedekind_domain
|
| gptkbp:application |
algebraic number theory
|
| gptkbp:characteristics |
gptkb:Monarch
|
| gptkbp:defines |
a ring in which every nonzero proper ideal factors into a product of prime ideals
|
| gptkbp:example |
the ring of integers
the ring of algebraic integers |
| gptkbp:has_property |
every nonzero prime ideal is maximal
the spectrum of a Dedekind ring is a finite union of points and curves every ideal can be uniquely factored into prime ideals the ring is a UFD if it is a Dedekind ring and has no nontrivial prime ideals every localization is a Dedekind ring finite integral extension has Krull dimension at most 1 integrally closed in its field of fractions the class group is finite the ideal class group is torsion the ring is Noetherian if it is a Dedekind ring the ring is a Dedekind ring if it is a Noetherian ring and every nonzero prime ideal is maximal. the ring of fractions is a Dedekind domain if the original ring is a Dedekind ring the ring is a PID if it is a Dedekind ring and every nonzero prime ideal is principal the ring is a Dedekind domain if it is a Noetherian integral domain of dimension 1 |
| https://www.w3.org/2000/01/rdf-schema#label |
Dedekind ring
|
| gptkbp:named_after |
gptkb:Richard_Dedekind
|
| gptkbp:related_concept |
gptkb:Dedekind_domain
Noetherian ring |
| gptkbp:type |
commutative ring
|