Zero divisor and unit elements with support 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 $ \mathbb{F} $ is a field, then the group ring $\mathbb{F}[G]$ contains no zero divisor and Kaplansky Unit Conjecture states that if $G $ is a torsion free group and $ \mathbb{F} $ is a field, then $\mathbb{F}[G]$ contains no non-trivial units. The support of an element $ \alpha= \sum_{x\in G}\alpha_xx$ in $\mathbb{F}[G] $, denoted by $supp(\alpha)$, is the set $ \{x \in G|\alpha_x\neq 0\} $. In this paper we study possible zero divisors and units with supports of size $ 4 $ in $\mathbb{F}[G]$. We prove that if $ \alpha, \beta $ are non-zero elements in $ \mathbb{F}[G] $ for a possible torsion free group $ G $ and an arbitrary field $ \mathbb{F} $ such that $ |supp(\alpha)|=4 $ and $ \alpha\beta=0 $, then $|supp(\beta)|\geq 7 $. In [J. Group Theory, $16$ $ (2013),$ no. $5$, $667$-$693$], it is proved that if $ \mathbb{F}=\mathbb{F}_2 $ is the field with two elements, $ G $ is a torsion free group and $ \alpha,\beta \in \mathbb{F}_2[G]\setminus \{0\}$ such that $|supp(\alpha)|=4 $ and $ \alpha\beta =0 $, then $|supp(\beta)|\geq 8$. We improve the latter result to $|supp(\beta)|\geq 9$. Also, concerning the Unit Conjecture, we prove that if $\mathsf{a}\mathsf{b}=1$ for some $\mathsf{a},\mathsf{b}\in \mathbb{F}[G]$ and $|supp(\mathsf{a})|=4$, then $|supp(\mathsf{b})|\geq 6$.