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 $\mathbb{F}$ 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 $\mathit{\text{supp}}(\alpha )$, 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 group algebras of torsion-free groups. We prove that if α, β are non-zero elements in $\mathbb{F}[G]$ for a possible torsion-free group G and an arbitrary field $\mathbb{F}$ such that $|\mathit{\text{supp}}(\alpha )|=4$ and $\alpha \beta =0$, then $|\mathit{\text{supp}}(\beta )|\ge 7$. 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 $\alpha ,\beta \in {\mathbb{F}}_2[G]\setminus \left\{0\right\}$ such that $|\mathit{\text{supp}}(\alpha )|=4$ and $\alpha \beta =0$, then $|\mathit{\text{supp}}(\beta )|\ge 8$. We improve the latter result to $|\mathit{\text{supp}}(\beta )|\ge 9$. Also, concerning the Unit Conjecture, we prove that if $ab=1$ for some $a,b\in \mathbb{F}[G]$ and $|\mathit{\text{supp}}(a)|=4$, then $|\mathit{\text{supp}}(b)|\ge 6$.

Keywords:

• Group ring
• Kaplansky unit conjecture
• Kaplansky zero divisor conjecture
• torsion-free group

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 20C07
• 16S34

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 $〈a,b|b{a}^m{b}^{-1}=a^n〉$.

Additional information

Funding

This work was supported by School of Mathematics, Institute for Research in Fundamental Sciences (IPM) [grant number 96050219] and the Center of Excellence for Mathematics, University of Isfahan.