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Free division rings of fractions of crossed products of groups with Conradian left-orders

“…Let x be additively indecomposable and assume that gx is additively decomposable. Then gx can be written as gx = a 1 + a 2 where a 1 , a 2 ⊳ gx and therefore x = g −1 a 1 + g −1 a 2 where cp(x) = cp(gx) and cp(a i ) = cp(g −1 a i ) for i = 1, 2 by [DGH,Proposition 4.8]. Thus, x is additively decomposable -a contradiction.…”

“…. , n is an atom by [DGH,Theorem 4.6]. Without loss of generality we just show cp(x 1 ) > 1 and assume cp(x 1 ) = 1.…”

“…Without loss of generality we just show cp(x 1 ) > 1 and assume cp(x 1 ) = 1. Then n > 1 and x ′ = x 1 x 2 is additively indecomposable because of [DGH,Lemma 4.5]. Furthermore, [DGH,Theorem 4.6] such that ρ(x ′ ) = cp(x 1 ) + cp(x 2 ).…”

“…Then n > 1 and x ′ = x 1 x 2 is additively indecomposable because of [DGH,Lemma 4.5]. Furthermore, [DGH,Theorem 4.6] such that ρ(x ′ ) = cp(x 1 ) + cp(x 2 ). On the other side x ′ is an atom since x 1 ∈ G by assumption.…”

“…We are mainly interested in the case when F [G, η, α] possesses a division ring of fractions D. By this we mean a skew field D which contains F [G, η, α] as a subring such that D is generated by F [G, η, α] as a division ring. In Section 5 we introduce the notion of freeness with respect to a Conradian left-order ≤ of G. Our first main result states that if D meets this freeness condition then any non-zero x ∈ D which is not a unit in F [G, η, α] possesses a "unique" normal form representation which can roughly be described as a local formal power series expansion over an archimedean ordered group where the coefficients are stricly simpler than x with respect to the complexity discussed in [D2,D3,DGH]. The archimedean ordered group corresponds to a factor group C/C ′ where (C ′ , C) is a convex jump of G with respect to the Conradian left-order ≤ and the coefficients are elements of the rational closure of F [C ′ , η, α] in D. From this we derive our next main result that any element of D can be comprehended as an endomorphism of the right F -vector space F ((G)) of all formal power series in G over F .…”