Free division rings of fractions of crossed products of groups with Conradian left-orders
- Let D be a division ring of fractions of a crossed product F[G, eta, alpha], 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, 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, eta, alpha] possesses a division ring of fraction which is free in this sense if and only if the rational closure of F[G, eta, alpha] in the endomorphism ring of the corresponding right F-vector space F((G)) is a skew field.
Author details: | Joachim GräterGND |
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DOI: | https://doi.org/10.1515/forum-2019-0264 |
ISSN: | 0933-7741 |
ISSN: | 1435-5337 |
Title of parent work (English): | Forum mathematicum |
Publisher: | De Gruyter |
Place of publishing: | Berlin |
Publication type: | Article |
Language: | English |
Date of first publication: | 2020/05/01 |
Publication year: | 2020 |
Release date: | 2023/01/02 |
Tag: | Conradian left-order; Hughes-free; crossed product; division ring of fractions; formal; group ring; locally indicable group; ordered group; power series |
Volume: | 32 |
Issue: | 3 |
Number of pages: | 34 |
First page: | 739 |
Last Page: | 772 |
Organizational units: | Mathematisch-Naturwissenschaftliche Fakultät / Institut für Mathematik |
DDC classification: | 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik |
Peer review: | Referiert |