Valuation ring - Wikipedia
WEBIn 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. Given a field F, if D is a subring of F such that either x or x−1 belongs to D for every nonzero x in F, then D is said to be a valuation ring for the field F or a place of F.
Section 10.50 (00I8): Valuation rings—The Stacks project
WEBLet A be a valuation ring. The totally ordered abelian group (Γ, ≥) of Lemma 10.50.12 is called the value group of the valuation ring A. The map v: A − {0} → Γ and also v: K∗ → Γ is called the valuation associated to A. The valuation ring A is called a …
WEBValuation Rings. Recall that a submonoid Γ+ of a group Γ defines an ordering on Γ, where. ≥ y iff x − y ∈ Γ+. The set of elements x such that x and −x both belong to Γ+ is a subgroup; dividing by this subgroup we find that in the quotient. ≥ y and y ≥ x implies x = y.
WEBAn integral domain A is called a discrete valuation ring if there is a discrete valuation v on the field of quotients of A so that A is the valuation ring of v. Example 9.2. (1) Let K = Q and for each prime p let v p: Q∗ → Z be the function given by v p(pka/b)=k if a,b are integers relatively prime to p. This is a discrete valuation with ...
WEBelds and valuation rings. that allows us to \localize" elds. We have seen how useful it can be to localize the ring A at a prime ideal p: this yields a discrete valuation ring Ap, a principal ideal domain with exactly one nonzero prime ideal, w.
WEB1.1 Absolute Values and Valuations. Definition. Let K be a field. An absolute value on K is a function | · | : K → R such that:
WEBIt is easy to verify that the valuation ring Ais a (commutative) ring, and even an integral domain (if xand yare nonzero then v(xy) = v(x) + v(y) 6= 1, so xy6= 0), and k is its fraction eld.
WEBA discrete valuation ring (DVR) is an integral domain that is the valuation ring of its fraction eld with respect to a discrete valuation; such a ring A cannot be a eld, since v(Frac A) = Z 6= Z 0 = v(A).
WEBValuation rings Theorem 1. Let R be a commutative domain with quotient field K. Then the following conditions on R are equivalent: (1) For any q ∈ K either q ∈ R or q 1 ∈ R. (2) Let a;b ∈ R be non-zero elements. Then either a|b or b|a (in R). (3) The set of ideals of R are linearly ordered. EXERCISE 1 - Prove Theorem 1.
Valuation Ring -- from Wolfram MathWorld
WEBAug 22, 2024 · Let (K,|·|) be a non-Archimedean field. Its valuation ring R is defined to be R= {x in K:|x|<=1}. The valuation ring has maximal ideal M= {x in K:|x|<1}, and the field R/M is called the residue field, class field, or field of digits.
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