Commutative algebra Valuation rings and Dedekind domains Concept
Discrete valuation ring
Summary
A discrete valuation ring is a principal ideal domain with exactly one nonzero prime ideal. It is the ring of elements of nonnegative value for a discrete valuation on its field of fractions. Discrete valuation rings are among the simplest rings that are not fields: their ideals form a single chain of powers of one element. They describe the local behaviour of curves at smooth points and of number rings at a prime.
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| Tag | MGM3CNJW9V |
|---|---|
| URI | m2://MGM3CNJW9V |
| Kind | concept |
| Writing | W0 stub |
| MSC | 13F30, 13H05, 13F10 |
| Specializes | Principal ideal domain, Valuation ring |
| Same as | |
| Changed | 2 November 2026 |
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Definitions
Used by
- Krull domain
- Characterizations of discrete valuation rings
- p-adic integer
- Cohen structure theorem
- Local field
- Divisor class group
- Ramification group