We are concerned in this paper with the ideal structure of group
rings of infinite
simple locally finite groups over fields of characteristic zero, and its
relation with
certain subgroups of the groups, called confined subgroups. The systematic study of
the ideals in these group rings was initiated by the second author in
[15], although
some results had been obtained previously (see [3, 1]).
Let G be an infinite simple
locally finite group and K a field of characteristic zero. It
is expected that in most
cases, the group ring KG will have the smallest possible number
of ideals, namely
three, (KG itself, {0} and the augmentation ideal), and this
has been verified in some
cases. In some interesting cases, however, the situation is different,
and there are more
ideals. We mention in particular the infinite alternating groups
[3] and the stable
special linear groups [9], in which the ideal lattice
has been completely determined.
The second author has conjectured that the presence of ideals in KG,
other than the
three unavoidable ones, is synonymous with the presence in the group of proper
confined subgroups. Here a subgroup H of a locally finite group
G is called confined,
if there exists a finite subgroup F of G such that
Hg∩F≠1 for all
g∈G. This amounts
to saying that F has no regular orbit in the permutation
representation of G on the cosets of H.