Kaplansky's idempotent conjecture for torsion-free groups in characteristic zero

Let $G$ be a torsion-free group and $k$ a field. The idempotent conjecture, commonly attributed to Kaplansky, asserts that the group ring $kG$ has no idempotents other than $0$ and $1$ . For $k=\mathbb C$ it follows from the Bass conjecture or from Baum-Connes for $G$ , and it was known for many classes of groups but not in general; it would also follow from Kaplansky's zero-divisor conjecture. In characteristic zero: if $G$ is torsion-free and $R$ a field (or domain) of characteristic zero, is every $e\in RG$ with $e^2=e$ equal to $0$ or $1$ ?

Result: Proved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Argument
Field: Group rings; idempotents
Posed by: Attributed to I. Kaplansky; the manuscript cites the formulation in Oinert and Wagner (2023), Problem 3
Year posed: —
Years open: —
Solved: 2026-09-24
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Lean-checked, statement unaudited
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 56 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Corollary 1.2(ii): for every torsion-free group $G$ and every commutative unital domain $R$ of characteristic zero, the only idempotents of $RG$ are $0$ and $1$ . Part (i): for torsion-free $G$ the canonical trace of an idempotent matrix over $\mathbb CG$ is the rank of its augmentation, so $\tau_*(K_0(\mathbb CG))=\mathbb Z$ . Both are deduced from the Bass trace theorem. Nothing is claimed in positive characteristic or about projections in $C^*_r(G)$ ; the release separately claims a torsion-free counterexample to the Kadison-Kaplansky projection conjecture there, which is consistent with this result.

What the AI did

Produced by an unreleased internal OpenAI model as part of the openai/math release (pinned commit adc7f12). The release README says results were produced by one fixed procedure averaging about three hours of ChatGPT Pro thinking compute each; this result is not among the README exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored as OpenAI with no human author named. A Lean formalization accompanies it (Comparator challenge BassTorsionFree).

Verification

No independent mathematician has checked this yet. Checked here: abstract, introduction, Corollary 1.2 and the torsion-free section of the TeX source, read against the idempotent conjecture as cited. Lean: Comparator challenge BassTorsionFree, declaration OAI.TorsionFreeBass.torsion_free_corollary. This challenge is not in lean/formalization.yaml; its JSON config and solution module exist at the pinned commit. Its statement was read here: for torsion-free $G$ , the trace of every idempotent matrix over $\mathbb C[G]$ equals the rank of its augmentation, the $K_0$ trace equals the augmentation rank with range $\mathbb Z$ , and for every commutative domain $R$ of characteristic zero every idempotent of $R[G]$ is $0$ or $1$ . The last clause is exactly the headline. Not rebuilt here. The paper notes its argument does not cover the reduced $C^*$ -algebra.

Kaplansky's idempotent conjecture for torsion-free groups in characteristic zero

What was actually shown

What the AI did

Verification

Sources

Discussion