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Algebra · Result 196

A counterexample to Kaplansky’s zero-divisor conjecture

1 paperLean

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.

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Papers 1

First page of A Torsion-Free Group Algebra with Zero Divisors

A Torsion-Free Group Algebra with Zero Divisors

OpenAI · September 23, 2026

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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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}
}

What was formalized in Lean

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 \(G\) and nonzero elements \(\alpha,\beta\in\mathbb F_2[G]\) with \(\alpha\beta=0\), giving a counterexample.

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

Comparator links

ResultComparator statement
Torsion-free group-algebra counterexampleTorsionFreeZeroDivisors.lean

Scope note on GitHub →