Browse › Algebra › 196 A counterexample to Kaplansky’s zero-divisor conjecture
A Torsion-Free Group Algebra with Zero Divisors
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Abstract
We disprove Kaplansky's zero-divisor conjecture by constructing a finitely presented torsion-free group G for which $\mathbb F_2[G]$ has nonzero zero divisors. The group admits a finite two-dimensional classifying space.
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Notes
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- acprk/openai-math-remarks, R14: Kaplansky zero-divisor counterexample over PG(2,16) (GitHub remark, undated; literature search stated up to 2026-10-09). Argues that the paper's construction and proof also work with projective-plane parameter q = 16 instead of q = 128, and reports a spectral-radius computation indicating the key lemma fails for q = 8; the remark states it assumes the q = 128 manuscript and was reviewed by AI sessions, not human peer review. [AI-generated per source]
Formalization
- OpenAI, A counterexample to Kaplansky’s zero-divisor conjecture — Lean scope note (lean/docs/196.md) (OpenAI math release, 6 October 2026). Describes the scope of the release's Lean formalization: a finitely presented torsion-free group with a finite two-dimensional K(G,1) and nonzero α, β ∈ F₂[G] with αβ = 0. [AI-generated per source]
- OpenAI, lean/OAI/Algebra/GroupRing/Main.lean (openai/math, commit adc7f12). The release's top-level Lean file, which proves
theorem main : MainTheoremfor a concrete group built from imported sampling, torsion-freeness, model and factor modules. - OpenAI, lean/ComparatorChallenges/TorsionFreeZeroDivisors.lean (openai/math, commit adc7f12). The comparator statement of the main theorem: ordinary torsion-freeness, Mathlib's
Group.IsFinitelyPresented, a finite 2-dimensional CW model with contractible cover, and nonzero zero divisors inMonoidAlgebra (ZMod 2) G. - Google DeepMind formal-conjectures, FormalConjectures/Wikipedia/Kaplansky.lean (formal-conjectures). Restates the zero-divisor conjecture as its negation, tags it as solved and cites this paper; proof bodies in the file are
sorryplaceholders linking to the OpenAI Lean proof. - mo271, feat: mark problems solved in the OpenAI math release and link formal proofs (formal-conjectures PR #6898, merged 2026-10-11). Lists the Kaplansky zero-divisor conjecture among the refuted statements, restates it as a negation and adds
counter_zero_divisor_conjecturewith a proof link. [family-level]
Possibly relevant literature (found by the agent, not checked)
- Pascal Schweitzer, On Zero Divisors with Small Support in Group Rings of Torsion-Free Groups (arXiv:1202.6645, 2012). Studies the zero-divisor conjecture through a reduction to the field with two elements, relevant to what a counterexample over F₂ such as the paper's main theorem implies for other fields.
- Alireza Abdollahi and Zahra Taheri, Kaplansky's zero divisor and unit conjectures on elements with supports of size 3 (arXiv:1612.00934, 2016). Proves lower bounds on the support size of β when αβ = 0 and |supp(α)| = 3 (at least 20 over F₂), constraining supports of factors like the paper's α and β.
- Moe Tabei, Least sizes of non-unique-product sets: the Promislow group and a Heisenberg-type candidate (arXiv:2607.18346, 2026). Studies minimal non-unique-product sets in torsion-free groups, the combinatorial obstruction the paper discusses alongside Rips–Segev, Steenbock and Mian–Siddique.
Digests and expositions
- Claude Opus 5.5 / Open Mathematics Courses, A torsion-free group algebra with zero divisors (Open Mathematics Courses, undated). Ten lessons explain the paper’s prescribed graph types, probabilistic estimates, planar separation, cone pictures and cancellation argument, with examples and exercises; the source states that independent review is incomplete. [AI-generated per source]
- AI Proved What?, Torsion-free, yet zero? Kaplansky’s zero-divisor conjecture (episode 196, October 2026). Video and transcript explain the parity mechanism and the separate nonvanishing and torsion-freeness arguments, illustrate a small toy construction, and report a source inspection of the Lean files without running the proof checker. [AI-generated per source]
News and blogs
- Azmat, Kaplansky Zero-Divisor Conjecture: Inside OpenAI’s Remarkable Counterexample (BinaryVerse AI, 10 October 2026; video version). Explains the manuscript’s characteristic-two cancellation, root separation and asphericity arguments and distinguishes its claim from the related unit conjecture.
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 196
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:A-Torsion-Free-Group-Algebra-with-Zero-Divisors-September-23-2026,
author = {{OpenAI}},
title = {{A Torsion-Free Group Algebra with Zero Divisors}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/A-Torsion-Free-Group-Algebra-with-Zero-Divisors-September-23-2026/paper.pdf}{OAI:A-Torsion-Free-Group-Algebra-with-Zero-Divisors-September-23-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/196.md. The Lean statements have not been checked by people against the paper's claims.
Family 196: A counterexample to Kaplansky’s zero-divisor conjecture
Constructs a finitely presented torsion-free group G whose group algebra $\mathbb F_2[G]$ has nonzero zero divisors, disproving Kaplansky's zero-divisor conjecture. The group has a finite two-dimensional classifying space.
This is the only paper in this family.