Group rings · torsion-free groups · noncommutative algebra · additive combinatorics

Kaplansky Zero-Divisor Conjecture

Collaboration beta

Can two nonzero finite linear combinations of elements of a torsion-free group ever multiply to zero in the group algebra over a field?

α, β \in K [G], α β = 0 \Rightarrow α = 0 or β = 0

Known results and sources

In a torsion-free group algebra, many coefficient products may cancel inside repeated multiplication fibers. Kaplansky's conjecture says complete cancellation cannot occur unless one factor was already zero.

Research problem

Exact mathematical statement

Let $K$ be a field and let $G$ be a torsion-free group. The group algebra $K[G]$ consists of finite sums $\sum_{g\in G} c_g g$ with coefficients in $K$ . Kaplansky's zero-divisor conjecture asserts that $K[G]$ is a domain in the zero-divisor sense:

\alpha,\beta\in K[G],\qquad \alpha\beta=0\quad\Longrightarrow\quad \alpha=0\ \text{or}\ \beta=0.

Thus no pair of nonzero finitely supported group-algebra elements should have zero product. The retained source states explicitly that this conjecture has not been proved and that it has produced no counterexample.

Problem infographic

Problem at a glance

A problem-first infographic defines a torsion-free group G, finitely supported group-algebra elements alpha and beta, and coefficient cancellation along product fibers; it contrasts the torsion-free assumption with the still-open conclusion that alpha beta equals zero only when alpha or beta equals zero. — Each coefficient of alpha beta is a sum over all support pairs with the same group product. Torsion-freeness forbids finite-order group elements, but whether it also prevents every nontrivial zero divisor remains open.

Current mathematical picture

Where work on Kaplansky Zero-Divisor Conjecture stands

Open conjecture

Selected route highlights from the current work. This is not yet a complete mathematical inventory.

Useful failureSource-reported limitation

The following claim is rejected or insufficient in the recorded route: A two-coset identity can contain all cancellation needed to close a rank-two local core without producing any further coset branch. The most concrete missing result is a tensor or subgroup-collapse theorem excluding both rooted 2 × 3 systems. Such an exclusion would close only the first rank-two three-fiber frontier; further stress ranks and support sizes would still require new arguments unless a broader compression theorem subsumes them.

Route status · Narrowed route

Main reductionA multidimensional collision-stress space is forced

The Jacobian of the product-fiber equations has a left-kernel collision-stress space whose dimension is at least three for a globally minimal pair.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeEliminate Pattern L by placing equations (37.9), (37.11), and (37.12) in correctly oriented twisted tensor products and closing the proportionality cycle.Task status · Work already reported in progress

Work mapped so far

Kaplansky Zero-Divisor Conjecture in numbers

2.7kretained lines of mathematical investigation2,673 in the current working snapshot

Argument development: 2,194 · 82%
Explored or eliminated routes: 94 · 4%
Computational analysis: 99 · 4%
Open obligations: 83 · 3%
Definitions and setup: 203 · 8%

9selected mapped statements1routes investigated3open questions2contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byGraham Higman (earliest sourced formulation), Irving Kaplansky (later published problem and conventional attribution)Proposed1940Status checkedAug 7, 2026

Source recordedAug 7, 2026

Recommended next task

Eliminate Pattern F by analyzing its inverse-stable four-cell cancellation as a tensor of rank at most two after the primitive identity pair is peeled.

Suggested move: Build one bimodule tensor model for all four terms, separate rank-one reduction from the genuine rank-two branch, and combine coprimality with identity-class matching to force a factor or coset collapse.

What would count as progress

Retain an exact proof or counterexample for the stated subproblem.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

13 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

1 recorded

Narrowed routeSource-reported limitation

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks

Eliminate Pattern F by analyzing its inverse-stable four-cell cancellation as a tensor of rank at most two after the primitive identity pair is peeled.Suggested move: Build one bimodule tensor model for all four terms, separate rank-one reduction from the genuine rank-two branch, and combine coprimality with identity-class matching to force a factor or coset collapse.

Ready to work on

Determine whether stress-guided multi-sandwich compression or higher-rank incidence geometry can handle cases not subsumed by the two rooted 2 × 3 patterns.Suggested move: Compute the exact support and kernel behavior of lambda mapping to beta r_lambda alpha on already bounded candidates, then formulate the first rank-three projective incidence lemma before considering a larger census.

Ready to work on

Eliminate Pattern L by placing equations (37.9), (37.11), and (37.12) in correctly oriented twisted tensor products and closing the proportionality cycle.Suggested move: Peel the common factors in the identity equation, identify each intersection subgroup and twisting automorphism, record both proportionality scalars, and compose them before making any subgroup-structure claim.

Work already reported in progress

Sourced mathematical context

The known mathematical landscape

Context collected Aug 7, 2026

Current statusOpen conjecture

Open in full generality: no authoritative accepted proof or counterexample is known for the assertion that K[G] has no nonzero zero divisors for every field K and torsion-free group G. It is proved for important classes including unique-product groups and, under the precise recent hypotheses, torsion-free 3-manifold and virtually compact special groups; these results do not cover every torsion-free group.

[3][4]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

classical special case
Authoritative summaryThe conjecture holds for unique-product groups by isolating a uniquely represented product in the support. Left- or right-orderable groups are important formalized examples, but torsion-free groups need not have the unique-product property.[3][7]
2026
Peer reviewedFisher and Sánchez-Peralta constructed division-ring embeddings in their stated virtually compact special and 3-manifold scopes, proving the zero-divisor conjecture for all torsion-free 3-manifold groups and stronger coefficient variants under their exact hypotheses.[4]
2024
Computational resultNielsen and Soelberg exhaustively analyzed finite subsets without unique products and proved |A| + |B| ≥ 16 for supports of a hypothetical zero-divisor pair, yielding corresponding lower bounds. This excludes small supports, not all counterexamples.[5]
1956–1957
Historical sourceKaplansky posed the zero-divisor assertion as Problem 6 in his 1956 conference contribution, published in 1957, helping establish the standard name and universal field/torsion-free-group formulation.[2][3]

Mathematical neighborhood

Related results and reusable starting points

Current focusKaplansky zero-divisor conjecture

Stronger or generalized formDivision-ring embedding conjecture

Embedding K[G] in a division ring immediately rules out zero divisors, but no-zero-divisors alone does not generally supply such an embedding. Coefficient-division-ring and crossed-product versions are also stronger than the canonical field statement.

[4]

Dependency or reductionStrong Atiyah conjecture

For torsion-free groups over C, appropriate strong Atiyah conjecture results yield a division closure that is a skew field and hence imply the zero-divisor assertion; this is a sufficient route, not an unconditional proof for all torsion-free groups.

[3][4]

Solved special caseUnique-product and orderable groups

Unique-product groups, including left- and right-orderable groups under the documented hypotheses, have zero-divisor-free group algebras over domains.

[3][7]

Solved special caseTorsion-free 3-manifold and virtually compact special groups

The conjecture is proved for torsion-free fundamental groups of 3-manifolds and for the paper's stated torsion-free virtually compact special scope through division-ring embeddings.

[4]

Related problemKaplansky unit conjecture

Kaplansky's unit conjecture is stronger in a different direction and is false, including in characteristic zero. Its counterexamples do not produce nonzero zero divisors and therefore do not refute this conjecture.

[3]

Logical consequenceKaplansky idempotent conjecture

A zero-divisor-free group algebra has no nontrivial idempotents because e(e−1)=0. Thus the zero-divisor conjecture implies the corresponding algebraic idempotent assertion, not conversely.

[2][3]

Formal and computational footholds

Existing statements, libraries, computations, and datasets that can shorten the next serious attempt.

formal statement · statement onlyFormal Conjectures: Kaplansky.lean
At exact repository commit c594af4ba42f58465253b8550545e0132959a78c, zero_divisor_conjecture states NoZeroDivisors (MonoidAlgebra K G) under field and torsion-free-group hypotheses but ends in sorry.
[6]
formal proof · source linked; not reproduced by ProofAtlasMathlib unique-product and ordered-group special cases
Mathlib's NoZeroDivisors module contains checked instances for monoid algebras of unique-product structures and orderable groups. These prove genuine special cases, not that every torsion-free group has the needed structure.
[7]
computation · not independently reproducedNielsen–Soelberg finite-support search
The peer-reviewed paper reports Magma and supercomputer enumeration establishing small-support exclusions. No public versioned code, machine-readable result set, and independently rerun certificate were verified in this pass.
[5]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

Formalization targetFormal comparison between the project's torsion-free group predicate and the algebraic hypotheses used by group-algebra instances.
Formalization targetA substantial formal theory of unique-product, locally indicable, virtually compact special, and 3-manifold groups with checked implications among the classes.
Formalization targetFormal division closures, affiliated operators, von Neumann dimensions, and the strong Atiyah route with exact coefficient hypotheses.
Formalization targetMachine-checkable certificates for exhaustive support searches and independently reproducible enumeration code.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

6 standing statements3 proposed statements3 open questions1 narrowed routes

Statements by mathematical role9 selected mapped statements

theorem candidate1 of 91
reduction3 of 93
lemma4 of 94
negative result1 of 91

Selected mathematical clusters3 mathematical clusters

Statements and reductionsClaims, implications, and derivations in the current map.17 displayed rows

retained route statementA torsion-free group should have a zero-divisor-free group algebra over every field.
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementMinimal product fibers are nontrivial matchingsintermediate
retained route statementA multidimensional collision-stress space is forcedintermediate
retained route statementA short stress exists and has rank at least twointermediate
retained route statementBalanced cancellation has full collision rankintermediate
retained route statementA matched two-coset return cannot closeintermediate
retained route statementOnly two rooted 2 × 3 systems remain at the first frontierintermediate
Recorded relationshipThe source material reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
DerivationThe current work reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed

Open questionsSpecific obligations that remain open in the current routes.3 displayed rows

Research targetEliminate Pattern L by placing equations (37.9), (37.11), and (37.12) in correctly oriented twisted tensor products and closing the proportionality cycle.in progress reported
Research targetEliminate Pattern F by analyzing its inverse-stable four-cell cancellation as a tensor of rank at most two after the primitive identity pair is peeled.open
Research targetDetermine whether stress-guided multi-sandwich compression or higher-rank incidence geometry can handle cases not subsumed by the two rooted 2 × 3 patterns.open

Explored routes and evidenceChallenges, computations, and approaches that have already narrowed the search.3 displayed rows · 1 route included

Useful failureSource-reported limitationreported failure
ComputationThe source reports embedded standard-library verifiers for the 22 connected rank-two color patterns and for the rooted 2 × 3 incidence census, together with expected exact counts and printed certificates.ProofAtlas did not execute either embedded verifier or inspect an independent archived run. Their results are retained only at the source's reported status; they are not promoted to computationally reproduced or formally verified evidence. · reported unreproduced
Narrowed routeSource-reported limitationThe following claim is rejected or insufficient in the recorded route: A two-coset identity can contain all cancellation needed to close a rank-two local core without producing any further coset branch. The most concrete missing result is a tensor or subgroup-collapse theorem excluding both rooted 2 × 3 systems. Such an exclusion would close only the first rank-two three-fiber frontier; further stress ranks and support sizes would still require new arguments unless a broader compression theorem subsumes them.

How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeEliminate Pattern L by placing equations (37.9), (37.11), and (37.12) in correctly oriented twisted tensor products and closing the proportionality cycle.

1 approach has already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Continue the mathematics

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Prepared starting pointEliminate Pattern F by analyzing its inverse-stable four-cell cancellation as a tensor of rank at most two after the primitive identity pair is peeled.

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Research contextPrepared context for any AI agent

Can two nonzero finite linear combinations of elements of a torsion-free group ever multiply to zero in the group algebra over a field?

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Current routes and known obstacles
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Sources and references9 cited works · next context review by Nov 7, 2026

The mathematical context was checked on Aug 7, 2026. Status can be refreshed sooner after a material result or claim.

1
Units in group ringsoriginal source · Graham Higman · University of Oxford · 1940 · DOI 10.5287/ora-bmo6o5bjx · accessed Aug 7, 2026
2
Problems in the theory of ringsoriginal source · Irving Kaplansky · National Academy of Sciences–National Research Council · 1957 · DOI 10.17226/20747 · accessed Aug 7, 2026
3
Units, zero-divisors and idempotents in rings graded by torsion-free groupspeer reviewed result · Johan Öinert · Journal of Group Theory · 2024 · DOI 10.1515/jgth-2023-0110 · accessed Aug 7, 2026
4
Division rings for group algebras of virtually compact special groups and 3-manifold groupspeer reviewed result · Sam P. Fisher, Pablo Sánchez-Peralta · Journal of Combinatorial Algebra · 2026 · DOI 10.4171/JCA/89 · accessed Aug 7, 2026
5
Small sets without unique products in torsion-free groupspeer reviewed result · Pace P. Nielsen, Lindsay Soelberg · Journal of Algebra and Its Applications · 2024 · DOI 10.1142/S0219498825500501 · accessed Aug 7, 2026
6
FormalConjectures/Wikipedia/Kaplansky.lean at c594af4ba42f58465253b8550545e0132959a78cformalization · The Formal Conjectures Authors · Google DeepMind · accessed Aug 7, 2026
7
Mathlib.Algebra.MonoidAlgebra.NoZeroDivisorsformalization · The Mathlib Contributors · Mathlib · accessed Aug 7, 2026
8
Kaplansky's conjecturesencyclopedia · Wikimedia Foundation · accessed Aug 7, 2026
9
List of unsolved problems in mathematicsencyclopedia · Wikimedia Foundation · accessed Aug 7, 2026

Important qualifications

The record covers the field-coefficient assertion that K[G] has no nonzero zero divisors for every field K and torsion-free group G. It does not conflate this with Kaplansky's unit or idempotent conjectures, the Kadison–Kaplansky conjecture, or the stronger division-ring embedding problem.
The proposed year 1940 denotes the earliest sourced formulation found in Graham Higman's thesis. Irving Kaplansky's 1956 conference problem, published in 1957, established the conventional attribution and title.
Solved group classes overlap substantially. The record gives representative unique-product, virtually compact special, and 3-manifold milestones rather than a complete lattice of locally indicable, amenable, one-relator, and related classes.
Coefficient-field, coefficient-domain, skew-field, and crossed-product variants are not treated as definitionally identical; stronger coefficient results retain their exact hypotheses.
Nielsen and Soelberg report an exhaustive Magma/supercomputer search, but this pass did not locate and rerun a public versioned code-and-output bundle, so the computation is not independently reproduced here.
The scoped formalization search covered the exact current Formal Conjectures file and current Mathlib no-zero-divisors module. It does not establish absence from other proof assistants or private developments.
No unreviewed source material, contributor claim, packet computation, or unpublished submission was inspected or used as external authority. This record grants no proof, novelty, review, acceptance, or publication authority.

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