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.
Verfasserangaben: | Joachim GräterGND |
---|---|
DOI: | https://doi.org/10.1515/forum-2019-0264 |
ISSN: | 0933-7741 |
ISSN: | 1435-5337 |
Titel des übergeordneten Werks (Englisch): | Forum mathematicum |
Verlag: | De Gruyter |
Verlagsort: | Berlin |
Publikationstyp: | Wissenschaftlicher Artikel |
Sprache: | Englisch |
Datum der Erstveröffentlichung: | 01.05.2020 |
Erscheinungsjahr: | 2020 |
Datum der Freischaltung: | 02.01.2023 |
Freies Schlagwort / Tag: | Conradian left-order; Hughes-free; crossed product; division ring of fractions; formal; group ring; locally indicable group; ordered group; power series |
Band: | 32 |
Ausgabe: | 3 |
Seitenanzahl: | 34 |
Erste Seite: | 739 |
Letzte Seite: | 772 |
Organisationseinheiten: | Mathematisch-Naturwissenschaftliche Fakultät / Institut für Mathematik |
DDC-Klassifikation: | 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik |
Peer Review: | Referiert |