Abstract
Kaplansky Zero Divisor Conjecture states that if G is a torsion-free group and is a field, then the group ring
contains no zero divisor and Kaplansky Unit Conjecture states that if G is a torsion-free group and
is a field, then
contains no non-trivial units. The support of an element
in
, denoted by
, is the set
. In this paper, we study possible zero divisors and units with supports of size 4 in group algebras of torsion-free groups. We prove that if α, β are non-zero elements in
for a possible torsion-free group G and an arbitrary field
such that
and
, then
. In [J. Group Theory, 16
no. 5, 667–693], it is proved that if
is the field with two elements, G is a torsion-free group and
such that
and
, then
. We improve the latter result to
. Also, concerning the Unit Conjecture, we prove that if
for some
and
, then
.
Note
Acknowledgements
The authors are grateful to the referee for his/her valuable suggestions and comments.
Notes
1 For integers m and n, the Baumslag–Solitar group BS(m, n) is the group given by the presentation .