What does valuation ring mean?
Definitions for valuation ring
val·u·a·tion ring
This dictionary definitions page includes all the possible meanings, example usage and translations of the word valuation ring.
Wikipedia
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 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. 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. 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, where ( A , m A ) {\displaystyle (A,{\mathfrak {m}}_{A})} dominates ( B , m B ) {\displaystyle (B,{\mathfrak {m}}_{B})} if A ⊃ B {\displaystyle A\supset B} and m A ∩ B = m B {\displaystyle {\mathfrak {m}}_{A}\cap B={\mathfrak {m}}_{B}} .Every local ring in a field K is dominated by some valuation ring of K. An integral domain whose localization at any prime ideal is a valuation ring is called a Prüfer domain.
Numerology
Chaldean Numerology
The numerical value of valuation ring in Chaldean Numerology is: 9
Pythagorean Numerology
The numerical value of valuation ring in Pythagorean Numerology is: 1
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"valuation ring." Definitions.net. STANDS4 LLC, 2024. Web. 28 Apr. 2024. <https://www.definitions.net/definition/valuation+ring>.
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