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 $\alpha ={\sum }_{x\in G}{\alpha }_{x}x$ in $\mathbb{F}[G]$, denoted by $supp(\alpha )$, is the set $\{x\in G|{\alpha }_x\ne 0\}$. 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 $|supp(\beta )|\ge 7$. 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 $|supp(\beta )|\ge 8$. We improve the latter result to $|supp(\beta )|\ge 9$. Also, concerning the Unit Conjecture, we prove that if $\mathsf{ab}=1$ for some $\mathsf{a},\mathsf{b}\in \mathbb{F}[G]$ and $|supp(\mathsf{a})|=4$, then $|supp(\mathsf{b})|\ge 6$.

97 pages