Theorems · Theorem · commutative algebra

Submodule.torsionBySet_isInternal

∀ {R : Type u_1} {M : Type u_2} [inst : CommRing R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] {ι : Type u_3}
  [inst_3 : DecidableEq ι] {S : Finset ι} {p : ι → Ideal R},
  ((↑S).Pairwise fun i j => p i ⊔ p j = ⊤) →
    Module.IsTorsionBySet R M ↑(⨅ i ∈ S, p i) → DirectSum.IsInternal fun i => Submodule.torsionBySet R M ↑(p ↑i)

If the p i are pairwise coprime, a ⨅ i, p i-torsion module is the internal direct sum of its p i-torsion submodules.

Defined in

Mathlib.Algebra.Module.Torsion.Basic

Cited by

2 results in Mathlib

Foundations

Depth 94 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModuleDecidableEq