Theorems · Theorem · ring theory

IsSemiprimaryRing.induction

∀ (R₀ : Type u_1) (R : Type u_2) (M : Type u) [inst : Ring R₀] [inst_1 : Ring R] [inst_2 : Module R₀ R]
  [inst_3 : AddCommGroup M] [inst_4 : Module R₀ M] [inst_5 : Module R M] [IsScalarTower R₀ R M] [IsSemiprimaryRing R]
  {P : (M : Type u) → [inst_8 : AddCommGroup M] → [Module R₀ M] → [Module R M] → Prop},
  (∀ (M : Type u) [inst_8 : AddCommGroup M] [inst_9 : Module R₀ M] [inst_10 : Module R M] [IsScalarTower R₀ R M]
      [IsSemisimpleModule R M], Module.IsTorsionBySet R M ↑(Ring.jacobson R) → P M) →
    (∀ (M : Type u) [inst_8 : AddCommGroup M] [inst_9 : Module R₀ M] [inst_10 : Module R M]
        [inst_11 : IsScalarTower R₀ R M],
        have N := Ring.jacobson R • ⊤;
        P ↥N → P (M ⧸ N) → P M) →
      P M

Defined in

Mathlib.RingTheory.HopkinsLevitzki

Cited by

2 results in Mathlib

Foundations

Depth 93 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingRingModuleAddCommGroupModuleModuleIsScalarTowerIsSemiprimaryRing