# Primitivity of group rings of non-elementary torsion-free hyperbolic groups

## TL;DR

This paper proves that group rings of non-elementary torsion-free hyperbolic groups over fields or countable domains are primitive, extending understanding of their algebraic structure.

## Contribution

It establishes the primitivity of group rings for a broad class of hyperbolic groups using recent theoretical results.

## Key findings

- Group rings over fields are primitive.
- Primitivity holds for countable domain coefficients.
- Extends algebraic understanding of hyperbolic groups.

## Abstract

We use a recent result of Alexander and Nishinaka to show that if $G$ is a non-elementary torsion-free hyperbolic group and $R$ is a countable domain, then the group ring $RG$ is primitive. This implies that the group ring $KG$ of any non-elementary torsion-free hyperbolic group $G$ over a field $K$ is primitive.