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 (Formula presented.) is a field, then the group ring (Formula presented.) contains no zero divisor and Kaplansky Unit Conjecture states that if G is a torsion-free group and (Formula presented.) is a field, then (Formula presented.) contains no non-trivial units. The support of an element (Formula presented.) in (Formula presented.), denoted by (Formula presented.), is the set (Formula presented.). 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 (Formula presented.) for a possible torsion-free group G and an arbitrary field (Formula presented.) such that (Formula presented.) and (Formula presented.), then (Formula presented.). In [J. Group Theory, 16 (Formula presented.) no. 5, 667–693], it is proved that if (Formula presented.) is the field with two elements, G is a torsion-free group and (Formula presented.) such that (Formula presented.) and (Formula presented.), then (Formula presented.). We improve the latter result to (Formula presented.). Also, concerning the Unit Conjecture, we prove that if (Formula presented.) for some (Formula presented.) and (Formula presented.), then (Formula presented.). © 2019, © 2019 Taylor & Francis Group, LLC.