# P50: Kaplansky zero-divisor conjecture

- ID: `P50`
- Reference: `kaplansky-zero-divisor-conjecture`
- Page: https://theoremdb.org/statements/P50
- Record maturity: Reviewed problem with recorded work

## Problem

If \(K\) is a field and \(G\) is a torsion-free group, then the group algebra \(K[G]\) has no nonzero zero divisors.

### Context

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

### Problem setup

- **Definition (A group).** A group is torsion-free when no element other than the identity has finite order.
- **Definition (A nonzero zero divisor).** A nonzero zero divisor is a nonzero algebra element a for which some nonzero b satisfies ab = 0.
- **Remark.** The conjecture asks whether torsion-free group structure prevents cancellation severe enough to create zero divisors in the associated algebra.

### What 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.

## Status

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](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (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.

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.

### Background and intake notes

- 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 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.
- The formulation and status were checked against the cited 2026 article on 2026-07-22.
- The related Kaplansky unit conjecture is false. This record concerns zero divisors only.

- Recorded example: The conjecture holds for many classes, including torsion-free three-manifold groups by the cited work.

### Open directions

- **Route 1** (reported): 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](#reference-1)

### Computational notes

- Finite-support searches can test specific groups and coefficient fields without covering arbitrary torsion-free groups.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `kaplansky-zero-divisor-conjecture`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>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 https://ems.press/journals/jca/articles/14297826
   - Also cited at Abstract and introduction
   - Also cited at Editorial research route recorded 2026-07-31
   - website; primary source; checked 2026-07-31
   - Source 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.
   - 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.