The universality of Hughes-free division rings
Tóm tắt
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.
Tài liệu tham khảo
Cohn, P.M.: Theory of General Division Rings. Encyclopedia of Mathematics and its Applications, vol. 57. Cambridge University Press, Cambridge (1995)
Malcev, A.I.: On the embedding of group algebras in division algebras. Doklady Akad. Nauk SSSR (N.S.) 60, 1499–1501 (1948). (Russian)
Malcolmson, P.: Determining homomorphisms to division rings. J. Algebra 64(2), 399–413 (1980)
Sánchez, J.: On division rings and tilting modules, Ph. D. Thesis, Universitat Autónoma de Barcelona (2008), www.tdx.cat/bitstream/handle/10803/3107/jss1de1.pdf