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Kaplansky Zero-Divisor Conjecture

The Kaplansky zero-divisor conjecture asserts that the group algebra K[G] of a torsion-free group G over a field K 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 p, there is a finitely presented torsion-free group G_p and nonzero alpha,beta in GF(p)[G_p] with alphabeta=0. 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. 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.

See also

Field Characteristic, Group Algebra, Kaplansky Direct-Finiteness Conjecture, Torsion-Free Group, Zero Divisor

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References

Ivanov, S. V. "Zero Divisors in Group Algebras of Torsion-Free Groups in Odd Characteristic." 13 Oct 2026. https://arxiv.org/abs/2610.17662.

Cite this as:

Weisstein, Eric W. "Kaplansky Zero-Divisor Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/KaplanskyZero-DivisorConjecture.html

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