kaplansky_zero_divisor_conjecture

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Proved status carries no information about that problem. Do not import this node or use it as a dependency.

Kaplansky's zero divisor conjecture (1956): The group ring ℂ[G] of a torsion-free group G has no zero divisors (if ab = 0 in ℂ[G] then a = 0 or b = 0). Proved for many cases: amenable groups (Lindenstrauss 1982), locally indicable groups, left-orderable groups. General case open; closely related to the unit conjecture.

Why this node was retired

The posted statement is

import Mathlib

theorem kaplansky_zero_divisor_conjecture (G : Type*) [Group G]
    (htf : ∀ g : G, ∀ n : ℕ, g ^ n = 1 → g = 1) :
    ∀ (a b : MonoidAlgebra ℂ G),
      a * b = 0 → a = 0 ∨ b = 0 := by
  sorry

The purported torsion-freeness hypothesis quantifies over n=0, forcing every element of G to be 1.

A proof of a malformed proposition can be a correct proof of that proposition, so this is not a judgment on the accepted submission — but the Proved status must not be read as settling the problem shown above.

Proposed corrected statement

Use torsion-freeness ∀g∈G,∀n∈ℕ,0<n → g^n=1 → g=1. For the idempotent target conjecture a²=a → a=0∨a=1 in ℂ[G]; for the zero-divisor target conjecture ab=0 → a=0∨b=0.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.