Kaplansky's zero divisor and unit conjectures on elements with supports of size $3$

TL;DR

This paper investigates the structure of zero divisors and units in group rings over torsion-free groups, establishing lower bounds on the support size of elements with support size 3, thus advancing the understanding of Kaplansky's conjectures.

Contribution

It proves new lower bounds on support sizes of zero divisors and units in group rings with support size 3, improving previous results for arbitrary fields and the field with two elements.

Findings

If $eta$ is a zero divisor partner of a support-3 element, then support size of $eta$ is at least 10 (20 over $ield_2$).

If $eta$ is a unit partner of a support-3 element, then support size of $eta$ is at least 9.

The results improve existing bounds related to Kaplansky's zero divisor and unit conjectures.

Abstract

Kaplansky's zero divisor conjecture (unit conjecture, respectively) states that for a torsion-free group $G$ and a field $\mathbb{F}$, the group ring $\mathbb{F}[G]$ has no zero divisors (has no unit with support of size greater than $1$). In this paper, we study possible zero divisors and units in $\mathbb{F}[G]$ whose supports have size $3$. For any field $\mathbb{F}$ and all torsion-free groups $G$, we prove that if $\alpha \beta=0$ for some non-zero $\alpha, \beta \in \mathbb{F}[G]$ such that $|supp(\alpha)|=3$, then $|supp(\beta)|\geq 10$. If $\mathbb{F}=\mathbb{F}_2$ is the field with 2 elements, the latter result can be improved so that $|supp(\beta)|\geq 20$. This improves a result in [J. Group Theory, 16 (2013), no. 5, 667-693]. Concerning the unit conjecture, we prove that if $\alpha \beta=1$ for some $\alpha, \beta \in \mathbb{F}[G]$ such that $|supp(\alpha)|=3$, then $|supp(\beta)|\geq 9$. The latter improves a part of a result in [Exp. Math., 24 (2015), 326-338] to arbitrary fields.