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

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 50 years and still remain open. 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 gradation. 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 have no non-homogeneous central unit, no non-trivial central zero-divisor and no non-trivial central idempotent. We also present generalizations of the classical group ring conjectures and show that it suffices to consider finitely generated torsion-free groups in order to solve them