Valuation Ring
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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 to D for every x in F, then D is said to be a valuation ring for the field F. Since F is in this case 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…mehr

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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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 to D for every x in F, then D is said to be a valuation ring for the field F. Since F is in this case 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. The valuation rings of a field are the maximal elements of the local subrings partially ordered by dominance, where (A,mathfrak{m}_A) dominates (B,mathfrak{m}_B) if A supset B and mathfrak{m}_A cap B = mathfrak{m}_B.