Zero divisor and unit elements with supports of size 4 in group algebras of torsion-free groups

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 .