Commutative algebra Valuation rings and Dedekind domains Concept
Dedekind domain
Summary
A Dedekind domain is an integral domain in which every nonzero proper ideal is, in a unique way, a product of prime ideals. Rings of integers of number fields are Dedekind domains, and so are all principal ideal domains. In such rings unique factorisation of elements may fail, but unique factorisation of ideals always holds. The ideal class group measures the gap between the two.
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| Tag | 7NPY2CN5KB |
|---|---|
| URI | m2://7NPY2CN5KB |
| Kind | concept |
| Writing | W0 stub |
| MSC | 13F05, 11R04 |
| Specializes | Integral domain, Noetherian ring |
| Same as | |
| Changed | 2 November 2026 |
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Definitions
Used by
- Splitting of prime ideals
- Class number
- Ideal class group
- Ramification
- Finiteness of the class number
- Fundamental identity for ramification and inertia
- Unique factorization of ideals in a Dedekind domain
- A Dedekind UFD is a PID