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 \({\mathcal {D}}_{E*G}\) for an arbitrary locally indicable group G is still an open question. Nevertheless, \({\mathcal {D}}_{E*G}\) exists, for example, if G is amenable or G is bi-orderable. In this paper we study, whether \({\mathcal {D}}_{E*G}\) 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 \({\mathcal {D}}_{E[G]}\) and it is universal. In Appendix we give a description of \({\mathcal {D}}_{E[G]}\) when G is a RFRS group.

设\(E*G\)是除环E和局部可指示群G的叉积。Hughes 证明了在\(E*G\) -同构之前，至多存在一个 Hughes-free Division \(E*G\) -ring。然而，对于任意局部可指示群G是否存在休斯自由除法\(E*G\) -ring \({\mathcal {D}}_{E*G}\)仍然是一个悬而未决的问题。尽管如此，\({\mathcal {D}}_{E*G}\) 还是存在的，例如，如果G是可服从的或G是可双序的。在本文中，我们研究是否\({\mathcal {D}}_{E*G}\)在某些情况下是分数的通用除法环。特别地，我们证明如果G是一个残差（局部可指示和服从的）群，那么存在\({\mathcal {D}}_{E[G]}\)并且它是普遍的。在附录中，当G是 RFRS 群时，我们给出了\({\mathcal {D}}_{E[G]}\) 的描述。