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C-Valuations and Normal C-Orderings

Published online by Cambridge University Press:  20 November 2018

M. Chacron
M. Chacron*
Affiliation:
Carleton University, Ottawa, Ontario
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Let D stand for a division ring (or skewfield), let G stand for an ordered abelian group with positive infinity adjoined, and let ω: D → G. We call to a valuation of D with value group G, if ω is an onto mapping from D to G such that

(i) ω(x) = ∞ if and only if x = 0,

(ii) ω(x1 + x2) = min(ω (x1), ω (x2)), and

(iii) ω (x1 x2) = ω (x1) + ω (x2).

Associated to the valuation ω are its valuation ring

R = ﹛x ∈ Dω(x) ≧ 0﹜,

its maximal ideal

J = ﹛x ∈ |ω(x) > 0﹜, and its residue division ring D = R/J.The invertible elements of the ring R are called valuation units. Clearly R and, hence, J are preserved under conjugation so that 1 + J is also preserved under conjugation. The latter is thus a normal subgroup of the multiplicative group Dm of D and hence, the quotient group D˙/1 + J makes sense (the residue group of ω). It enlarges in a natural way the residue division ring D (0 excluded, and addition “forgotten“).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 41 , Issue 1 , 01 February 1989 , pp. 14 - 67
Copyright
Copyright © Canadian Mathematical Society 1989

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