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Free division rings of fractions of crossed products of groups with Conradian left-orders
Forum Mathematicum ( IF 1.0 ) Pub Date : 2020-05-01 , DOI: 10.1515/forum-2019-0264
Joachim Gräter 1
Affiliation  

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:

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,
更新日期:2020-05-01
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