  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.
