abelian_radical_iff_solvable_is_abelian

English

Over Noetherian rings, the radical is abelian iff every solvable ideal is abelian.

Русский

Над числами безграничного множества, абелевость радикала эквивалентна тому, что каждый разрешимый идеал абелев.

LaTeX

$$$(IsNoetherian\; R\; L) : IsLieAbelian (radical\ R\ L) \iff \forall I : LieIdeal\ R\ L, IsSolvable I \Rightarrow IsLieAbelian I$$$

Lean4

theorem abelian_radical_iff_solvable_is_abelian [IsNoetherian R L] :
    IsLieAbelian (radical R L) ↔ ∀ I : LieIdeal R L, IsSolvable I → IsLieAbelian I :=
  by
  constructor
  · rintro h₁ I h₂
    rw [LieIdeal.solvable_iff_le_radical] at h₂
    exact (LieIdeal.inclusion_injective h₂).isLieAbelian h₁
  · intro h; apply h; infer_instance

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