Result 196, Algebra

A counterexample to Kaplansky’s zero-divisor conjecture

Constructs a finitely presented torsion-free group G whose group algebra F2[G]\mathbb F_2[G] has nonzero zero divisors, disproving Kaplansky's zero-divisor conjecture. The group has a finite two-dimensional classifying space.

Lean formalization Disproof or counterexample

The bigger picture

Why it matters

An unreviewed manuscript reports two nonzero expressions whose product is zero in an algebra built from a torsion-free group. This would overturn Kaplansky's conjectured link between a group's structure and the behavior of multiplication.

What changes?

The example is a torsion-free group: no element except the identity has finite order. Its group algebra over the two-element field consists of finite formal sums of group elements, with coefficients added modulo two and multiplication extending the group operation. Two nonzero such sums multiply to zero. The group has finitely many generators and defining relations, and a finite two-dimensional classifying space, a geometric model encoding the group using finitely many cells of dimension at most two.

What does that help mathematicians do?

If correct, the construction rules out torsion-freeness alone as a sufficient condition for a group algebra to have no zero divisors. It also shows that imposing a finite presentation or a finite two-dimensional classifying space does not rescue that assertion over the two-element field. Researchers seeking positive results would therefore need additional restrictions. The example does not establish analogous counterexamples over other fields.

Are there practical applications?

The immediate value is foundational: the claimed example would clarify which structural assumptions can support reliable multiplication rules in group algebras. Its finite geometric model also places the obstruction within a tightly constrained setting connecting algebra and topology. The supplied material describes a mathematical counterexample, not a practical algorithm or technological application.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

A Torsion-Free Group Algebra with Zero Divisors

September 23, 2026 26 pages Main result formalized in Lean

We disprove Kaplansky's zero-divisor conjecture by constructing a finitely presented torsion-free group G for which F2[G]\mathbb F_2[G] has nonzero zero divisors. The group admits a finite two-dimensional classifying space.

PDF LaTeX source
Cite (BibTeX) 
@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's note on what the formalization covers, from lean/docs/196.md.

A counterexample to Kaplansky’s zero-divisor conjecture

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

Kaplansky's zero-divisor conjecture asserts that the group algebra of a torsion-free group over a field has no zero divisors. The formalized result constructs a finitely presented torsion-free group GG and nonzero elements α,β∈F2[G]\alpha,\beta\in\mathbb F_2[G] with αβ=0\alpha\beta=0, giving a counterexample.

The same group admits a finite two-dimensional classifying space K(G,1)K(G,1).

Comparator links

Result Comparator statement
Torsion-free group-algebra counterexample TorsionFreeZeroDivisors.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.