Kaplansky Zero-Divisor Conjecture

The Kaplansky zero-divisor conjecture asserts that the group algebra of a torsion-free group over a field has no nonzero zero divisors. This is distinct from conjectures about units or idempotents in group algebras.

Ivanov (2026) reports that, for each odd prime number , there is a finitely presented torsion-free group and nonzero with . Together with an external characteristic-two construction, Ivanov (2026) concludes that counterexamples exist over every field of positive field characteristic. The proof imports substantial parts of that earlier construction, which have not been independently verified here. The characteristic-zero case remains open.

Ivanov (2026) reports substantial Claude assistance with the signed-cancellation argument, the construction, computations, and review of the inherited proof. The author takes responsibility for the work.