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...
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.
