Math Conjectures

Kaplansky Zero-Divisor Conjecture

OPENMajorConjectureProposed c. 1940 · Full conjecture

Canonical statement

If KK is a field and GG is a torsion-free group, then the group algebra K[G]K[G] has no nonzero zero divisors: for all a,b∈K[G]a,b\in K[G], ab=0ab=0 implies a=0a=0 or b=0b=0.
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If \(K\) is a field and \(G\) is a torsion-free group, then the group algebra \(K[G]\) has no nonzero zero divisors: for all \(a,b\in K[G]\), \(ab=0\) implies \(a=0\) or \(b=0\).
ALG-RING-004:[Kaplansky1957Problems][Linnell1993DivisionRings][FisherSanchezPeralta2026]
Algebra and representation theoryGeometry and topology

Notes

Kaplansky's zero-divisor conjecture asserts that if KK is a field and GG a torsion-free group, then the group algebra K[G]K[G] is a domain: ab=0ab=0 forces a=0a=0 or b=0b=0. Torsion-freeness is necessary, since an element gg of finite order n>1n>1 gives (1−g)(1+g+⋯+gn−1)=0(1-g)(1+g+\cdots+g^{n-1})=0. The conjecture circulated among Kaplansky's group-ring problems long before appearing in print, so the customary date of about 1940 is only conventional; a printed source is his 1957 problem list [Kaplansky1957Problems].

The conjecture is known for large classes of groups. It holds for locally indicable groups, and one fertile route has been to embed K[G]K[G] into a division ring, a theme developed in Linnell's work on group von Neumann algebras [Linnell1993DivisionRings]. Recently this division-ring approach was carried out for group algebras of virtually compact special groups and of torsion-free 33-manifold groups, establishing the conjecture for the latter class [FisherSanchezPeralta2026].

For arbitrary torsion-free groups the question is open in both directions: no counterexample is known, and no general method rules one out.

References (3)

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