Abstract
AbstractLet $$E*G$$
E
∗
G
be a crossed product of a division ring E and a locally indicable group G. Hughes showed that up to $$E*G$$
E
∗
G
-isomorphism, there exists at most one Hughes-free division $$E*G$$
E
∗
G
-ring. However, the existence of a Hughes-free division $$E*G$$
E
∗
G
-ring $${\mathcal {D}}_{E*G}$$
D
E
∗
G
for an arbitrary locally indicable group G is still an open question. Nevertheless, $${\mathcal {D}}_{E*G}$$
D
E
∗
G
exists, for example, if G is amenable or G is bi-orderable. In this paper we study, whether $${\mathcal {D}}_{E*G}$$
D
E
∗
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 $${\mathcal {D}}_{E[G]}$$
D
E
[
G
]
and it is universal. In Appendix we give a description of $${\mathcal {D}}_{E[G]}$$
D
E
[
G
]
when G is a RFRS group.
Publisher
Springer Science and Business Media LLC
Subject
General Physics and Astronomy,General Mathematics
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