Algebra·OpenAI Math·Result 196·Lean formalisation listed
A counterexample to Kaplansky’s zero-divisor conjecture
One of the 372 results in OpenAI's public mathematics catalogue, filed under algebra: 1 preprint, the latest dated 23 September 2026. This page quotes OpenAI's own summary of the claim, links the paper and abstract, and embeds the animation.
Also known as Kaplansky zero-divisor conjecture; Kaplansky's zero divisor conjecture; zero-divisor conjecture for group rings
In one line: OpenAI reports a counterexample to Kaplansky’s zero-divisor conjecture (algebra) in a preprint dated 23 September 2026. It has not been peer reviewed. The channel's animation, Kaplansky Zero-Divisor Conjecture: OpenAI Claims a Counterexample, is on YouTube.
What OpenAI claims
Constructs a finitely presented torsion-free group G whose group algebra 𝔽2[G] has nonzero zero divisors, disproving Kaplansky's zero-divisor conjecture. The group has a finite two-dimensional classifying space. (Lean)
Paper
A Torsion-Free Group Algebra with Zero Divisors
OpenAI's abstract We disprove Kaplansky's zero-divisor conjecture by constructing a finitely presented torsion-free group G for which 𝔽2[G] has nonzero zero divisors. The group admits a finite two-dimensional classifying space.
Status
Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.
Quick answers
- Is it proven?
- Not independently yet. OpenAI posted a preprint on 23 September 2026; it has not been peer reviewed, so treat the result as OpenAI's claim until other mathematicians have checked it.
- What exactly does OpenAI claim?
- Constructs a finitely presented torsion-free group G whose group algebra 𝔽2[G] has nonzero zero divisors, disproving Kaplansky's zero-divisor conjecture. The group has a finite two-dimensional classifying space. (Lean)
- Where is the proof?
- In the paper, A Torsion-Free Group Algebra with Zero Divisors (PDF).
- Is there a video?
- Yes, on YouTube (0:21).
Also in
- Chapter 34 of 84 in Every OpenAI Math Result, Part 2 of 4: PCP for PPAD, Birkhoff Conjecture and 82 More · starts at 11:26 · watch from there on YouTube →
- Chapter 4 of 10 in Serre's Multiplicity Conjecture, Kaplansky Zero-Divisor Conjecture and 8 More · starts at 1:03 · watch from there on YouTube →
The video
Kaplansky Zero-Divisor Conjecture: OpenAI Claims a Counterexample · Short · 0:21 · description, sources and transcript →
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