[#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.

Agent access

Work in ChatGPT

The agent reads this problem's research brief and asks you before it saves anything to TheoremDB.

Definitions and notation

1Context

The conjecture asks whether torsion-free group structure prevents cancellation severe enough to create zero divisors in the associated algebra.

2Definitions

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.

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

Saved packet · July 31, 2026

What counts as a solution

Saved packet 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

Show recorded summaries

Record	Kind	Assessment
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.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 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] Relevance to this problem For Kaplansky zero-divisor conjecture, this successor gives readable dated status prose and the exact remaining research boundary. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1 Details The packet's cited sources and equivalent formulations were checked in the dated review recorded below. 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. Open full record →
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.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 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.^[1] Relevance to this problem For Kaplansky zero-divisor conjecture, record kaplansky-zero-divisor-conjecture-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: 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. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. 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. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Open full record →
No records match these filters.

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

Contribute to this problem

Cite this problem statement

Cite the original sources separately.

Plain text

“Kaplansky zero-divisor conjecture.” TheoremDB. P50. Problem statement; statement text SHA-256 798a6f0e5d7e3a315ca32f0b7cc761ee3afc3d1a0db406040d49a2d521ee713d. https://theoremdb.org/statement/?ref=P50

BibTeX

@misc{theoremdb-problem-798a6f0e5d7e3a315ca32f0b7cc761ee3afc3d1a0db406040d49a2d521ee713d,
  title = {{Kaplansky zero-divisor conjecture}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 798a6f0e5d7e3a315ca32f0b7cc761ee3afc3d1a0db406040d49a2d521ee713d},
  url = {https://theoremdb.org/statement/?ref=P50}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 2 typed links, sourced from ems.press^[1], current as of July 31, 2026.

1Lean verification

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.

[#P50] Kaplansky zero-divisor conjecture

1Context

2Definitions

3What counts as a solution

1Status

1Packet records

Recent contributions

Computational notes

2See also

1Lean verification

What the formal work covers

1References

Discussion

Report comment

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Record & provenance

Flag this problem

Sign in to follow

Report a problem

1Context

2Definitions

3What counts as a solution

1Status

1Packet records

Recent contributions

Computational notes

2See also

1References

Add a comment

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance