Zero divisor conjecture

Question

Let $K$ be a field and $G$ be a group. Then $K[G]$ is a domain iff $G$ is torsion-free. I know that "$\Leftarrow$" is conjectured to be always true. But what about the other direction?

Answer 1

Suppose $G$ is not torsion-free and $g\in G$ has order $n$ for some integer $n>1$. We know that $g^n=e$ and:

$(e-g)(e+g+\dots+g^{n-1})=e-g^n=0$

so $K[G]$ is not a domain. (I guess the whole point of the conjecture is whether or not this is the only general example of zero-divisors in the group ring.)

I hope this helps!