# Units, zero-divisors and idempotents in rings graded by torsion-free groups

## TL;DR

This paper extends classical problems about units, zero-divisors, and idempotents from group rings of torsion-free groups to more general rings graded by torsion-free groups, providing partial solutions and new insights.

## Contribution

It introduces generalized problems in the context of rings graded by torsion-free groups and offers partial solutions for unique product groups, advancing the understanding of these classical issues.

## Key findings

- Partial solutions for rings graded by unique product groups.
- Rings graded by torsion-free groups are indecomposable.
- Such rings have no non-trivial central zero-divisors or non-homogeneous central units.

## Abstract

The three famous problems concerning units, zero-divisors and idempotents in group rings of torsion-free groups, commonly attributed to I. Kaplansky, have been around for more than 60 years and still remain open in characteristic zero. In this article, we introduce the corresponding problems in the considerably more general context of arbitrary rings graded by torsion-free groups. For natural reasons, we will restrict our attention to rings without non-trivial homogeneous zero-divisors with respect to the given grading. We provide a partial solution to the extended problems by solving them for rings graded by unique product groups. We also show that the extended problems exhibit the same (potential) hierarchy as the classical problems for group rings. Furthermore, a ring which is graded by an arbitrary torsion-free group is shown to be indecomposable, and to have no non-trivial central zero-divisor and no non-homogeneous central unit. We also present generalizations of the classical group ring conjectures.