Discrete Valuation Ring

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:

  1. R is a local principal ideal domain, and not a field.
  2. R is a valuation ring with a value group isomorphic to the integers under addition.
  3. R is a local Dedekind domain and not a field.
  4. R is a noetherian local ring with Krull dimension one, and the maximal ideal of R is principal.
  5. R is an integrally closed noetherian local ring with Krull dimension one.
  6. R is a principal ideal domain with a unique non-zero prime ideal.
  7. R is a principal ideal domain with a unique irreducible element (up to multiplication by units).
  8. R is a unique factorization domain with a unique irreducible element (up to multiplication by units).
  9. 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.
  10. 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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