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Kaplansky’s idempotent conjecture in characteristic zero

Kaplansky’s characteristic-zero idempotent question asks whether every scalar idempotent of R[G] is 0 or 1 when G is torsion-free and R is a commutative characteristic-zero domain. Matrix idempotents and positive-characteristic coefficients are separate settings.

Why it matters and what counts as progress

Why it matters: It asks whether the absence of group torsion forces this basic ring-theoretic rigidity throughout characteristic zero.

What counts as progress: Exclude every nontrivial scalar idempotent for torsion-free groups over characteristic-zero domains. Matrix idempotents and positive characteristic are separate targets.

How the problem developed

  1. 1957 · Background [1][2]

    Irving Kaplansky Kaplansky published a list of ring-theory problems that shaped the conjectures now commonly associated with his name; later sources note that an original citation for the specific idempotent conjecture is elusive.

  2. 24 Sept 2026 · Theorem claim [2][3]

    OpenAI Released a Bass-trace manuscript deriving that every scalar idempotent in R[G] is 0 or 1 for torsion-free discrete G and commutative characteristic-zero domains R.

AI contribution

Sources

  1. K- and L-theory of Group Rings, International Congress of Mathematicians 2010 proceedings
  2. OpenAI, Bass trace manuscript and idempotent corollary
  3. Bass manuscript torsion-free corollary
  4. The Bass trace conjecture and the characteristic-zero Kaplansky idempotent conjecture
  5. Lean proof of the torsion-free corollary
  6. Comparator theorem configuration
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