# When is a group ring an integral domain

If $R$ is an integral domain (I am having $\mathbb{Z}$ or a field in mind) and $G$ a (not necessarily finite) group then
we can form the group ring $R(G)$.

Note that if $g^{n+1} = e$ then $(e-g)(e+g\ldots + g^n) = e - g^{n+1} = 0$. This means
if $G$ has torsion then $R(G)$ always has zero-divisors.

What about the inverse? So if $G$ is torsion-free does that imply $R(G)$ having no zero-divisors.

By embedding $R$ into its field of fractions, we may as well assume that $R$ is a field. But then this is precisely Kaplansky’s zero-divisor conjecture. This is a hard problem which is still open (2014). It has only been solved for certain classes of groups. If $R$ has characteristic $0$ then it suffices to treat $R=\mathbb{C}$ and there analytical methods are available. A reference is