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

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 $\mathbb{G}$.

The results are found to be known. Also, it has not published yet