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alternative definition of Krull valuation
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(Definition)
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Let $G$ be an abelian totally ordered group, denoted additively. We adjoin to $G$ a new element $\infty$ such that $g < +\infty$ , for all $g\in G$ and we extend the addition on $G_{\infty} = G \cup \{+\infty\}$ by declaring $g + (+\infty) = (+\infty) + (+\infty) = +\infty$ .
Definition 1 Let $R$ be an unital ring, a valuation of $R$ with values in $G$ is a function from $R$ to $G_{\infty}$ such that , for all $x, y \in R$ :
1) $v(xy)= v(x)+v(y)$ ,
2) $v(x+y) \geq \min \{v(x),v(y)\}$ ,
3) $v(x)=+\infty$ iff $v(x)=0$ .
Remarks a) The condition 1) means that $v$ is a homomorhism of $R\smallsetminus\{0\}$ with multiplication in the group $G$ . In particular, $v(1)=0$ and $v(-x)=v(x)$ , for all $x\in G$ . If $x$ is invertible then $0=v(1)=v(xx^{-1})=v(x)+v(x^{-1})$ , so $v(x^{-1}) = -v(x)$ .
b) If 3) is replaced by the weaker condition $v(0)=+\infty$ then the set $P = v^{-1}\{+\infty\}$ is a prime ideal of $R$ and $v$ is induced on the integral domain $R/P$ .
c) In particular, conditions 1) and 3) imply that $R$ is an integral domain and let $K$ be its quotient field. There is a unique valuation of $K$ with values in $G$ that extends $v$ , namely $v(x/y)=v(x)-v(y)$ , for all $x\in R$ and $y\in R\smallsetminus\{0\}$ .
d) The element $v(x)$ is sometimes denoted by $vx$ .
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"alternative definition of Krull valuation" is owned by polarbear.
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Cross-references: quotient field, integral domain, prime ideal, invertible, multiplication, iff, function, valuation, unital ring, addition, group, totally ordered, abelian
This is version 7 of alternative definition of Krull valuation, born on 2007-05-02, modified 2007-06-01.
Object id is 9322, canonical name is AlternativeDefinitionOfValuation2.
Accessed 901 times total.
Classification:
| AMS MSC: | 11R99 (Number theory :: Algebraic number theory: global fields :: Miscellaneous) | | | 12J20 (Field theory and polynomials :: Topological fields :: General valuation theory) | | | 13A18 (Commutative rings and algebras :: General commutative ring theory :: Valuations and their generalizations) | | | 13F30 (Commutative rings and algebras :: Arithmetic rings and other special rings :: Valuation rings) |
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Pending Errata and Addenda
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