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“).