Math Conjectures

Kaplansky Direct-Finiteness Conjecture

OPENMajorConjectureProposed c. 1940 · Full conjecture

Canonical statement

For every field KK, every group GG, and all a,b∈K[G]a,b\in K[G], if ab=1ab=1, then ba=1ba=1.
View source LaTeX
For every field \(K\), every group \(G\), and all \(a,b\in K[G]\), if \(ab=1\), then \(ba=1\).
ALG-RING-006:[Kaplansky1969FieldsRings][ElekSzabo2004Sofic][CeccheriniSilbersteinCoornaert2025][OpenAI2026TenAdvances]
Algebra and representation theoryGeometry and topology

Notes

Kaplansky's direct-finiteness conjecture asserts that group algebras are directly finite: for every field KK, every group GG, and all a,b∈K[G]a,b\in K[G], the relation ab=1ab=1 forces ba=1ba=1. Unlike the zero-divisor and idempotent conjectures, no torsion-freeness is assumed. The problem circulated from around 1940, a conventional date, and a printed formulation appears in Kaplansky's Fields and Rings [Kaplansky1969FieldsRings].

Elek and Szabó proved direct finiteness of K[G]K[G], for arbitrary KK, whenever GG is sofic [ElekSzabo2004Sofic]. An August 1, 2026 manuscript announces an explicit non-sofic group [OpenAI2026TenAdvances], but that claim is still awaiting independent verification and the manuscript does not settle direct finiteness for the announced group.

The conjecture remains open: a full resolution must treat arbitrary groups directly, and the existence of a non-sofic group alone does not supply either a counterexample or a proof.

References (4)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.