Ore condition for group algebras

DefinitionShift of the lamplighter base group#✓fully proved

For a natural number \(p\), the shift is the automorphism \(f \mapsto f(\cdot - 1)\) of the base group \(\bigoplus_{i \in \mathbb{Z}} \mathbb{Z}/p\), the additive group of finitely supported functions \(\mathbb{Z} \to \mathbb{Z}/p\), viewed as a multiplicative group.

noncomputable def shift (p : ℕ) : MulAut (Multiplicative (ℤ →₀ ZMod p)) :=
  AddEquiv.toMultiplicative (Finsupp.domCongr (Equiv.addRight (1 : ℤ)))

Used by 2

Used byLamplighter group · Kourovka Problem 12.47: some lamplighter group algebra is Ore

AbbreviationLamplighter group#✓fully proved

For a natural number \(p\), the lamplighter group \(\mathbb{Z}/p \wr \mathbb{Z}\) is the semidirect product \(\left(\bigoplus_{i \in \mathbb{Z}} \mathbb{Z}/p\right) \rtimes \mathbb{Z}\), in which \(n \in \mathbb{Z}\) acts on the base group by the \(n\)-th power of the shift \(f \mapsto f(\cdot - 1)\).

abbrev Lamplighter (p : ℕ) : Type :=
  Multiplicative (ℤ →₀ ZMod p) ⋊[zpowersHom (MulAut (Multiplicative (ℤ →₀ ZMod p))) (shift p)]
    Multiplicative ℤ

Uses 1 · Used by 1

Statement usesShift of the lamplighter base group

Used byKourovka Problem 12.47: some lamplighter group algebra is Ore

DefinitionNon-zero-divisor#✓fully proved

An element \(s\) of a ring \(R\) is a non-zero-divisor when it is neither a left nor a right zero divisor: for every \(z \in R\), \(zs = 0\) implies \(z = 0\) and \(sz = 0\) implies \(z = 0\).

def IsNonZeroDivisor {R : Type*} [Ring R] (s : R) : Prop :=
  (∀ z : R, z * s = 0 → z = 0) ∧ ∀ z : R, s * z = 0 → z = 0

Used by 1

Used byRight Ore ring

DefinitionRight Ore ring#✓fully proved

A ring \(R\) satisfies the right Ore condition for its non-zero-divisors, so that it has a classical right ring of quotients, when for all \(s, t \in R\) with \(s\) a non-zero-divisor there are \(x, y \in R\) with \(x\) a non-zero-divisor and \(sy = tx\).

For group algebras the left and right conditions agree.

def IsOreRing (R : Type*) [Ring R] : Prop :=
  ∀ s t : R, IsNonZeroDivisor s → ∃ x y : R, IsNonZeroDivisor x ∧ s * y = t * x

Uses 1 · Used by 1

Statement usesNon-zero-divisor

Used byKourovka Problem 12.47: some lamplighter group algebra is Ore

DefinitionKourovka Problem 12.47: some lamplighter group algebra is Ore#✓fully proved

There are a field \(k\) and a prime \(p\) such that the group algebra \(k[\mathbb{Z}/p \wr \mathbb{Z}]\) of the lamplighter group \(\mathbb{Z}/p \wr \mathbb{Z}\) satisfies the right Ore condition for its non-zero-divisors, that is, has a classical right ring of quotients.

Kourovka 12.47 (P. A. Linnell) asks this for a given field and prime; the existential form makes its refutation cover every field and every prime.

def lamplighterGroupAlgebraOre : Prop :=
  ∃ (k : Type) (_ : Field k) (p : ℕ), p.Prime ∧ IsOreRing (MonoidAlgebra k (Lamplighter p))

Uses 3 · Used by 1

Statement usesRight Ore ring · Lamplighter group · Shift of the lamplighter base group

Used byLamplighter group algebras are not Ore

TheoremLamplighter group algebras are not Ore#●ready to prove

For every field \(k\) and every prime \(p\), the group algebra \(k[\mathbb{Z}/p \wr \mathbb{Z}]\) of the lamplighter group \(\mathbb{Z}/p \wr \mathbb{Z}\) does not satisfy the right Ore condition for its non-zero-divisors: there are \(s\) and \(t\) in it, with \(s\) a non-zero-divisor, such that \(sy = tx\) holds for no \(x\) and \(y\) with \(x\) a non-zero-divisor.

Resolved false: Linnell, Lück and Schick (Theorem 3, with W. Dicks's argument) prove that \(kG\) is not an Ore ring for every field \(k\) and every group \(G\) containing \(H \wr \mathbb{Z}\) with \(H \ne 1\) finite, in particular for \(G = \mathbb{Z}/p \wr \mathbb{Z}\) and every prime \(p\); the Kourovka Archive answers 12.47 "No, it does not".

public theorem_wanted lamplighterGroupAlgebraOre_refuted : ¬ lamplighterGroupAlgebraOre

Sources

• E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, 21st issue, arXiv:1401.0300v49 (2026), Archive of Solved Problems, Problem 12.47 (P. A. Linnell); resolution: P. A. Linnell, W. Lück and T. Schick, The Ore condition, affiliated operators, and the lamplighter group, in: High-dimensional manifold topology (Trieste, 2001), World Scientific, 2003, 315-321, Theorem 3: kG is not an Ore ring for every field k and every group G containing H ≀ ℤ with H ≠ 1 finite; Kourovka Archive, Problem 12.47, http://him-lueck.uni-bonn.de/data/lamplighter.pdf.

Uses 1

Statement usesKourovka Problem 12.47: some lamplighter group algebra is Ore

5 fully proved1 ready to prove

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