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

Abstract

Abstract Let D be a division ring of fractions of a crossed product F ⁢ [ G , η , α ] {F[G,\eta,\alpha]} , where F is a skew field and G is a group with Conradian left-order ≤ {\leq} . For D we introduce the notion of freeness with respect to ≤ {\leq} and show that D is free in this sense if and only if D can canonically be embedded into the endomorphism ring of the right F-vector space F ⁢ ( ( G ) ) {F((G))} of all formal power series in G over F with respect to ≤ {\leq} . From this we obtain that all division rings of fractions of F ⁢ [ G , η , α ] {F[G,\eta,\alpha]} which are free with respect to at least one Conradian left-order of G are isomorphic and that they are free with respect to any Conradian left-order of G. Moreover, F ⁢ [ G , η , α ] {F[G,\eta,\alpha]} possesses a division ring of fraction which is free in this sense if and only if the rational closure of F ⁢ [ G , η , α ] {F[G,\eta,\alpha]} in the endomorphism ring of the corresponding right F-vector space F ⁢ ( ( G ) ) {F((G))} is a skew field.