Valuation ring
In abstract algebra, a valuation ring is an integral domain D such that for every element x of its field of fractions F, at least one of x or x −1 belongs to D. Given a field F, if D is a subring of F such that either x or x −1 belongs toD for every nonzero x in F, then D is said to be a valuation ring for the field F or a place of F. Since F in this case is indeed the field of fractions of D, a valuation ring for a field is a valuation ring. Another way to characterize the valuation rings of a field F is that valuation rings D of F have F as their field of fractions, and their ideals are totally ordered by inclusion; or equivalently their principal ideals are totally ordered by inclusion. In particular, every valuation ring is a local ring.
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Absolute value (algebra)Adele ringArchimedean propertyBranch pointBézout domainCenter (valuation ring)Combinatorics: The Rota WayCommutative algebraCompletion of a ringDimension theory (algebra)Discrete valuation ringDivisibility (ring theory)Glossary of algebraic geometryGlossary of commutative algebraGlossary of ring theoryHasse–Arf theoremHyperreal numberIgusa zeta functionIntegerIntegral elementIntegrally closed domainLocal ringLocal uniformizationNoetherian ringOswald TeichmüllerProper morphismPrüfer domainReal closed ringRestricted power seriesRigid analytic spaceRing (mathematics)Saunders Mac LaneSerial moduleStrassmann's theoremTannakian formalismV-topologyValuation (algebra)Valuation domainValuative criterionZariski–Riemann space
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Valuation ring
In abstract algebra, a valuation ring is an integral domain D such that for every element x of its field of fractions F, at least one of x or x −1 belongs to D. Given a field F, if D is a subring of F such that either x or x −1 belongs toD for every nonzero x in F, then D is said to be a valuation ring for the field F or a place of F. Since F in this case is indeed the field of fractions of D, a valuation ring for a field is a valuation ring. Another way to characterize the valuation rings of a field F is that valuation rings D of F have F as their field of fractions, and their ideals are totally ordered by inclusion; or equivalently their principal ideals are totally ordered by inclusion. In particular, every valuation ring is a local ring.
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In abstract algebra, a valuati ...... ing is called a Prüfer domain.
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In abstract algebra, a valuati ...... aluation ring is a local ring.
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Valuation ring
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