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.

中文翻译：

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具有康弧左阶的群的交叉积的分数的自由除环

摘要 令 D 为交叉积 F ⁢ [ G , η , α ] {F[G,\eta,\alpha]} 的分数的除环，其中 F 是斜场，G 是具有左康弧度的群订单 ≤ {\leq} 。对于 D，我们引入了关于 ≤ {\leq} 的自由度的概念，并证明 D 在这个意义上是自由的当且仅当 D 可以规范地嵌入到右 F 向量空间 F ⁢ ( ( G ) ) {F((G))} 在 G 上关于 ≤ {\leq} 的所有形式幂级数。由此我们得到 F ⁢ [ G , η , α ] {F[G,\eta,\alpha]} 的分数的所有除环，它们相对于 G 的至少一个康弧左阶是自由的，是同构的，并且对于 G 的任何康弧左阶，它们是自由的。此外， F ⁢ [ G , η , α ] {F[G,\eta,