Kaplansky's Conjectures
The zero-divisor conjecture If G is torsion-free, then the group algebra K[G] has no non-trivial zero divisors.
From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.
Cite
@misc{cairn-kaplansky,
title = {Kaplansky's Conjectures},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/kaplansky}},
year = {2026},
note = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
} Also: CITATION.cff · Atom feed of results
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Current state
No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.
The problem
The question
zero_divisor_conjecture. The zero-divisor conjecture
If G is torsion-free, then the group algebra K[G] has no non-trivial zero divisors.
idempotent_conjecture. The idempotent conjecture
If G is torsion-free, then K[G] has no non-trivial idempotents.
Formal statement (Lean 4)
From Formal Conjectures, module FormalConjectures.Wikipedia.Kaplansky (2 statements).
theorem zero_divisor_conjecture : NoZeroDivisors (MonoidAlgebra K G)theorem idempotent_conjecture (a : MonoidAlgebra K G) (h : IsIdempotentElem a) :
a = 0 ∨ a = 1What counts as progress
- A Lean proof of one of the statements above, pinned as the claim's formal statement.
- Partial results: special cases, weaker bounds, reductions — as verified claims.
- Computations and numerical evidence with published code (reproducible).
- Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
- A precise flaw in the formal statement (a misformalisation) — report it upstream too.
References
Throughout, "torsion-free" means that the identity is the only element of finite order (Monoid.IsTorsionFree). This is weaker than Mathlib's IsMulTorsionFree, which asks for uniqueness of roots and fails for the Promislow group below.
Source and licence
Imported from Formal Conjectures (Wikipedia), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.