Theorems · Theorem · commutative algebra

IsNoetherian.induction

∀ {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] [IsNoetherian R M]
  {P : Submodule R M → Prop}, (∀ (I : Submodule R M), (∀ J > I, P J) → P I) → ∀ (I : Submodule R M), P I

If ∀ I > J, P I implies P J, then P holds for all submodules.

Defined in

Mathlib.RingTheory.Noetherian.Basic

Cited by

2 results in Mathlib

Foundations

Depth 84 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringAddCommMonoidModuleIsNoetherian