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

TL;DR

This paper investigates zero divisors and units with support size 4 in group algebras of torsion free groups, establishing lower bounds on the support size of related elements and improving existing results.

Contribution

It proves new lower bounds on support sizes of zero divisors and units with support 4 in group algebras, refining previous bounds and advancing the understanding of Kaplansky's conjectures.

Findings

If $eta$ is a zero divisor with support 4, then $|supp(eta)| \\geq 7$ in general.

For the field with two elements, the support of $eta$ must be at least 9.

If a product of elements with support 4 equals 1, the support of the inverse is at least 6.

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 $|supp(\mathsf{b})|\geq 6$.