Kaplansky's conjectures

Remarks

The conjectures are linked: the unit conjecture implies the zero divisor conjecture, and the zero divisor conjecture implies the idempotent conjecture, since an idempotent $e$ satisfies $e(1 - e) = 0$. The hypothesis that $G$ is torsion-free is needed: if $g \ne 1$ has finite order $n$, then $(1 - g)(1 + g + \cdots + g^{n-1}) = 1 - g^n = 0$. The unit conjecture goes back to the 1940 thesis of Graham Higman. In 2021 Giles Gardam disproved the unit conjecture over the field with two elements, with a nontrivial unit in the group ring of the fundamental group of the Hantzsche–Wendt manifold. All three conjectures hold for groups with the unique product property, and the zero divisor conjecture holds for torsion-free elementary amenable groups. The zero divisor and idempotent conjectures are open in general. Kaplansky also made conjectures in other fields, on Banach algebras and on quadratic forms; they are separate questions.