Localization (mathematics) | Commutative algebra

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 domain whose maximal ideal is principal, and not a field. 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 Noetherian, not a field, and every nonzero fractional ideal of R is irreducible in the sense that it cannot be written as a finite intersection of fractional ideals properly containing it. 10. * There is some discrete valuation ν on the field of fractions K of R such that R = {0} {x K : ν(x) ≥ 0}. (Wikipedia).

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