A counterexample to Kaplansky's zero-divisor conjecture at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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A counterexample to Kaplansky's zero-divisor conjecture

at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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Difficulty:	🧠🧠🧠🧠🧠	Ages:	13 - ∞
Skills:	multiplying things	Levels:	1
Category:	Algebra	Lean version:	YES! ✔
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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.

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LEVEL 1 A Torsion-Free Group Algebra with Zero Divisors

released 2026-09-23 | 1 theorem · 7 lemmas · 11 proofs · 12,309 words | PLAY LEVEL 1 » (pdf)

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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