Zero divisors with small supports in group algebras of torsion-free groups over a field

TL;DR

This paper investigates zero divisors with minimal support in group algebras over torsion-free groups, establishing specific relations among group elements when the support size is three.

Contribution

It proves that certain relations must hold among group elements when a zero divisor with support size three exists in the algebra.

Findings

Zero divisors with support size three impose specific relations among group elements.

Conditions for zero divisors are characterized in torsion-free group algebras.

The structure of such zero divisors is constrained by the group's properties.

Abstract

For any field $\mathbb{F}$ and all torison-free group $\mathbb{G}$, we prove that if $ab = 0$ for some non-zero $a, b \in \mathbb{F}[\mathbb{G}]$ such that $|supp(a)|$ $= 3$ and $a = 1 + \alpha_{1}g_{1} + \alpha_{2}g_{2}$, then $g_{1}, g_{2}$ satisfies certain relations in $\mathbb{G}$.