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

Passman, Donald S. The algebraic structure of group rings. Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977.

For a summary of the known results, see MO/79559.

Placing the integers $\{1,2,\ldots,n\}$ on a circle ( for $n>1$) in some special order

Is the complement of an injective continuous map $\mathbb{R}\to \mathbb{R}^2$ with closed image necessarily disconnected?

Closure of $\left\{\frac{\tan{n}}{n} | \, n \in \mathbb{N}\right\}$

Good introductory books on homological algebra

Proof for why a matrix multiplied by its transpose is positive semidefinite

Proof of the unbounded-ness of $\sum_{n\geq 1}\frac{1}{n}\sin\frac{x}{n}$

Proving the inequality $\frac{\log (1)}{1!}+\frac{\log ^2(2)}{2!}+\frac{\log^3(3)}{3!}+\cdots> \frac{\pi }{4}$

Taking the automorphism group of a group is not functorial.

Where Fermat's last theorem fails

Is there an elementary expression for every real sequence?

Why would I define Alexander–Spanier cohomology?