The universality of Hughes-free division rings

Andrei Jaikin-Zapirain ^[1]
1. [1] Universidad Autónoma de Madrid
  
  Universidad Autónoma de Madrid
  
  Madrid, España
Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 27, Nº. 4, 2021
Idioma: inglés
DOI: 10.1007/s00029-021-00691-w
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Resumen
- Let E∗G be a crossed product of a division ring E and a locally indicable group G. Hughes showed that up to E∗G-isomorphism, there exists at most one Hughes-free division E∗G-ring. However, the existence of a Hughes-free division E∗G-ring DE∗G for an arbitrary locally indicable group G is still an open question. Nevertheless, DE∗G exists, for example, if G is amenable or G is bi-orderable. In this paper we study, whether DE∗G is the universal division ring of fractions in some of these cases. In particular, we show that if G is a residually-(locally indicable and amenable) group, then there exists DE[G] and it is universal. In Appendix we give a description of DE[G] when G is a RFRS group.