A Note on Group Rings of Certain Torsion-Free Groups

Abstract

As a step towards characterizing ID-groups (i.e., groups G such that, for every ring R without zero-divisors, the group ring RG has no zero-divisors), Rudin and Schneider defined Ω-groups, a possibly wider class than that of right-orderable groups, and proved that if every non-trivial finitely generated subgroup of a group G has a non-trivial H-group as an epimorphic image, then G is an ID-group. We prove that such groups are even Ω-groups and obtain the analogous result for right-orderable groups.

Discretely ordered groups

“…Recall that a group G is locally indicable if every finitely generated nontrivial subgroup of G has an infinite cyclic quotient. Such groups are right-orderable, as was shown by Burns and Hale [1972]. On the other hand, a right-orderable group need not be locally indicable as was shown by Bergman [1991].…”

A property of groups and the Cauchy–Davenport theorem

“…Burns and Hale [3] proved that every locally indicable group G is right orderable, i.e., a total order c can be defined on G so that a c b implies ac c bc for every c A G. In particular, free groups, onerelator torsion-free groups, nilpotent torsion-free groups and their free and direct products are locally indicable and, by Theorem 1.1 (b), possess the property Cðd; 1Þ, which gives some evidence in support of Conjecture 1.2.…”

Conjugacy in lattice-ordered groups and right ordered groups

“…, y n ) = 1 is torsion-free. By a result of S. D. Brodsky ([5], Corollary 2.3), H is locally indicable and so right orderable [6]. Now let (G, ≤) be any right ordered group.…”