Direct finiteness of positive-characteristic group rings with torsion
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Is every group ring directly finite: does ab=1 imply ba=1? The studied test is scalar elements over a finite characteristic-two field and a finitely presented group with odd-prime torsion.
Why it matters and what counts as progress
Why it matters: A scalar counterexample would go beyond a matrix defect and the classes already known to be directly finite, including sofic groups.
What counts as progress: Explicit scalar a,b with ab=1 and ba≠1, or a theorem ruling them out; the field must be specified rather than automatically identified with F₂.
How the problem developed
- 2004 · Theorem [1]
Gábor Elek and Endre Szabó Proved direct finiteness for group rings of sofic groups over arbitrary fields, a positive result under a group-theoretic restriction.
OpenAI The manuscript reports a characteristic-two counterexample to Kaplansky direct finiteness using a group with odd-prime torsion.
AI contribution
A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Characteristic Two
There are a finite field K of characteristic two, a finitely presented group G containing odd-prime torsion, and scalar a,b in K[G] with ab=1 but ba≠1. The manuscript gives a terminating prescription for the witness; it does not assert K=F₂.
Sources
- Gábor Elek and Endre Szabó, Sofic groups and direct finiteness
- OpenAI, A Counterexample to Kaplansky’s Direct Finiteness Conjecture in Characteristic Two
- OpenAI Lean documentation for group-ring direct finiteness
- A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Characteristic Two
- Characteristic-two direct-finiteness manuscript
- Direct-finiteness Lean theorem
- Formalization scope
- Torsion-free direct-finiteness companion
Website timeline
Publication dates and editorial changes.