Open problem, Algebra, posed 1957

Kaplansky's conjectures on group rings

Partially solved

Let $K$ be a field and $G$ a torsion-free group. The unit conjecture states that every unit of the group ring $K[G]$ is trivial, i.e. of the form $kg$ for $k \in K \setminus \{0\}$ and $g \in G$ . The zero-divisor conjecture states that $K[G]$ has no non-trivial zero divisors. The idempotent conjecture states that $K[G]$ has no idempotents other than $0$ and $1$ . The unit conjecture implies the zero-divisor conjecture, which implies the idempotent conjecture.

Research frontier as of 2026

As of 2026, the unit conjecture is fully settled as false in every characteristic (Gardam 2021, Murray 2021, Gardam 2023). The zero-divisor and idempotent conjectures remain open for general torsion-free groups, though proven for elementary amenable groups, torsion-free hyperbolic groups (via the Farrell–Jones conjecture), and groups satisfying the Atiyah conjecture; ongoing work also searches for the minimal support size of Gardam-type units and studies whether analogous non-trivial units exist for unique-product-like groups.

Best known results

The unit conjecture is false over every field: an explicit counterexample of support size $21$ exists in $\mathbb{F}_2[P]$ (Gardam 2021), extended to every $\mathbb{F}_p[P]$ (Murray 2021) and to $\mathbb{C}[P]$ (Gardam 2023).
The zero-divisor conjecture holds for all elementary amenable groups and, in characteristic zero, for all groups satisfying the Atiyah conjecture; the idempotent conjecture holds for all groups satisfying the Farrell–Jones conjecture, including all hyperbolic and CAT(0) groups.

Tools and where they stop

Tool	Achieved	Where it stops
SAT solving and computer search over word balls	Found the first explicit counterexample to the unit conjecture by encoding the unit equation over $\mathbb{F}_2[P]$ as a Boolean satisfiability instance	Computational search cost grows too fast with support size and group complexity to test general torsion-free groups, especially over characteristic-zero fields
Ore domain and Atiyah/Farrell–Jones conjecture machinery	Proves the zero-divisor and idempotent conjectures for elementary amenable, hyperbolic, and CAT(0) groups by relating $K[G]$ to Ore localization and $L^2$ -cohomology	Requires structural hypotheses (amenability, hyperbolicity, or the Farrell–Jones/Atiyah conjectures) that are unproven or false outside their known classes of groups

Open questions

Does the zero-divisor conjecture hold for every torsion-free group, or does a counterexample exist as for the unit conjecture?
What is the minimum support size of a non-trivial unit in $\mathbb{F}_2[G]$ over any torsion-free group $G$ ; is $21$ optimal for the Promislow group?

References

Giles Gardam (2021). A counterexample to the unit conjecture for group rings · DOI:10.4007/annals.2021.194.3.9 · arXiv:2102.11818
Alan D. Murray (2021). Zero divisors and idempotents in group rings · arXiv:2106.02147
Giles Gardam (2023). Non-trivial units of complex group rings · arXiv:2312.05240 [preprint, not peer-reviewed]