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

  • 1 Institut für Mathematik, Universität Potsdam, Karl-Liebknecht-Straße 24–25, 14476, Potsdam, Germany
Joachim Gräter
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  • Institut für Mathematik, Universität Potsdam, Karl-Liebknecht-Straße 24–25, 14476, Potsdam, OT Golm, Germany
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Let D be a division ring of fractions of a crossed product F[G,η,α], where F is a skew field and G is a group with Conradian left-order . For D we introduce the notion of freeness with respect to 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)) of all formal power series in G over F with respect to . From this we obtain that all division rings of fractions of F[G,η,α] 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] possesses a division ring of fraction which is free in this sense if and only if the rational closure of F[G,η,α] in the endomorphism ring of the corresponding right F-vector space F((G)) is a skew field.

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