Geodesic
C-Valuations and Normal C-Orderings
Canadian journal of mathematics, Tome 41 (1989) no. 1, pp. 14-67

Voir la notice de l'article provenant de la source Cambridge University Press

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“). 
Chacron, M. C-Valuations and Normal C-Orderings. Canadian journal of mathematics, Tome 41 (1989) no. 1, pp. 14-67. doi: 10.4153/CJM-1989-002-6
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