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In abstract algebra, a valuation ring is an integral domain D such that for every non-zero element x of its field of fractions F, at least one of x or x−1 belongs to D.
Examples, Definitions & Dominance and integral closure
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ring valuation displaystyle field ideal mathbb mathfrak ordered local place maximal totally called prime integral ideals domain fractions every rings
| Subject | Predicate | Object | Confidence | Src |
|---|---|---|---|---|
| Valuation ring | is a | integral domain D such that for every non-zero element x of its field of fractions F | 0.90 | text |
| Valuation ring | is a | local ring.The valuation rings of a field are the maximal elements of the set of the local subrings in the field partially ordered by dominance or refinement | 0.90 | text |
| Valuation ring | is a | local domain | 0.90 | text |
| Valuation ring | related to Definitions | There | 0.60 | section |
| Valuation ring | related to Definitions | For | 0.60 | section |
| Valuation ring | related to Definitions | The | 0.60 | section |
| Valuation ring | related to Dominance and integral closure | The | 0.60 | section |
| Valuation ring | related to Dominance and integral closure | This | 0.60 | section |
| Valuation ring | related to Dominance and integral closure | Since | 0.60 | section |
| Valuation ring | related to Dominance and integral closure | D/M | 0.60 | section |
| Valuation ring | related to Dominance and integral closure | In | 0.60 | section |
| Valuation ring | related to Dominance and integral closure | Every | 0.60 | section |
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