Kaplansky’s idempotent conjecture for integral quandle rings

The conjecture asks whether, for a finite Latin quandle $Q$, the integral quandle ring $\mathbb{Z}[Q]$ has only trivial idempotents. The retrieved sources do not identify its proposer or date.

Known results

• The conjecture was previously known for $R_3$ and $R_5$.
• Integral quandle rings of free quandles have only trivial idempotents.
• Certain finite-type coverings and knot quandles provide infinitely many or nontrivial idempotents, so the broad formulation requires hypotheses.
• Earlier work computed idempotents for several specific quandle rings, including $\mathbb{Z}[R_3]$, $\mathbb{Z}[R_4]$, and $\mathbb{Z}[\mathrm{Cs}(4)]$.

September 2026 special-case resolution

A September 2026 paper claims the conjecture for Takasaki quandles, including dihedral quandles, and for medial commutative quandles. It introduces Fourier analysis for Alexander quandles and proves necessity of a previously known sufficient condition for counterexamples. This is a substantial advance, but not a resolution for all integral quandle rings.

Current status (as of September 2026): The conjecture is claimed for Takasaki and medial commutative quandles, while the general case remains open and the new claims are unverified.