In abstract algebra, a **discrete valuation ring** (**DVR**) is a principal ideal domain (PID) with exactly one non-zero maximal ideal.

This means a DVR is an integral domain *R* which satisfies any one of the following equivalent conditions:

*R*is a local principal ideal domain, and not a field.*R*is a valuation ring with a value group isomorphic to the integers under addition.*R*is a local Dedekind domain and not a field.*R*is a noetherian local ring with Krull dimension one, and the maximal ideal of*R*is principal.*R*is an integrally closed noetherian local ring with Krull dimension one.*R*is a principal ideal domain with a unique non-zero prime ideal.*R*is a principal ideal domain with a unique irreducible element (up to multiplication by units).*R*is a unique factorization domain with a unique irreducible element (up to multiplication by units).*R*is not a field, and every nonzero fractional ideal of*R*is irreducible in the sense that it cannot be written as finite intersection of fractional ideals properly containing it.- There is some Dedekind valuation ν on the field of fractions
*K*of*R*, such that*R*={*x*:*x*in*K*, ν(*x*) ≥ 0}.

Read more about Discrete Valuation Ring: Examples, Uniformizing Parameter, Topology

### Other articles related to "discrete valuation ring, ring, valuation":

**Discrete Valuation Ring**- Topology

... Every

**discrete valuation ring**, being a local

**ring**, carries a natural topology and is a topological

**ring**... Intuitively an element z is "small" and "close to 0" iff its

**valuation**ν(z) is large ... Examples of complete DVRs include the

**ring**of p-adic integers and the

**ring**of formal power series over any field ...

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