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
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