[#P50] Kaplansky zero-divisor conjecture
Problem. If \(K\) is a field and \(G\) is a torsion-free group, then the group algebra \(K[G]\) has no nonzero zero divisors.
1Context
The conjecture asks whether torsion-free group structure prevents cancellation severe enough to create zero divisors in the associated algebra.
2Problem setup
Definition 1 (A group). A group is torsion-free when no element other than the identity has finite order.
Definition 2 (A nonzero zero divisor). A nonzero zero divisor is a nonzero algebra element a for which some nonzero b satisfies ab = 0.
Remark 1. The conjecture asks whether torsion-free group structure prevents cancellation severe enough to create zero divisors in the associated algebra.
3What counts as a solution
- Prove the absence of nonzero zero divisors for every torsion-free group and field, or give a torsion-free group, a field, and explicit nonzero group-algebra elements whose product is zero.
1Status
Current status (Dated status and exact unresolved remainder). Unresolved in this packet after the dated source check. Strongest checked result: The zero-divisor conclusion holds for torsion-free 3-manifold groups and substantial virtually compact special and previously known elementary amenable classes. Exact unresolved remainder: For every field K and every torsion-free group G, prove that K[G] has no nonzero zero divisors, or exhibit a counterexample.[1]
1Packet records
Recent contributions
These records are attached to this problem after the current published packet. Each badge shows its current verification or packet-review step.
Notes and companion material
Original intake status. The cited 2026 article states that Kaplansky's zero-divisor conjecture remains wide open in general. The source and public status were checked on 2026-07-31. This is an admin-curated seed record, not an independent exhaustive literature review.
- The related Kaplansky unit conjecture is false. This record concerns zero divisors only.
Recorded example 1. The conjecture holds for many classes, including torsion-free three-manifold groups by the cited work.
Computational notes
- Finite-support searches can test specific groups and coefficient fields without covering arbitrary torsion-free groups.
2See also
How to cite
TheoremDB contributors, “Kaplansky zero-divisor conjecture,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/kaplansky-zero-divisor-conjectureThis page as plain text: kaplansky-zero-divisor-conjecture.md
This problem includes 2 records joined by 2 typed links, sourced from ems.press[1], current as of July 31, 2026.
1References
- Packet source. Sam P. Fisher and Pablo Sánchez-Peralta, Division rings for group algebras of virtually compact special groups and 3-manifold groups, Journal of Combinatorial Algebra 10 (2026), 153-193. DOI 10.4171/JCA/89. Sam P. Fisher and Pablo Sánchez-Peralta, Journal of Combinatorial Algebra 10 (2026), introduction. ↗website · primary source · checked 2026-07-31Source use: original summary.The cited 2026 article states that Kaplansky's zero-divisor conjecture remains wide open in general. The source and public status were checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.Also cited at Abstract and introduction.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.States the general status and proves the result for torsion-free 3-manifold groups and related classes.Source named by the research packet.
An original CC0 restatement prepared by TheoremDB maintainers.