Theorems · Theorem · commutative algebra

Submodule.le_of_isLocalized_span

∀ {R : Type u_1} {M : Type u_2} [inst : CommSemiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (s : Set R),
  Ideal.span s = ⊤ →
    ∀ (Mₚ : ↑s → Type u_5) [inst_3 : (r : ↑s) → AddCommMonoid (Mₚ r)] [inst_4 : (r : ↑s) → Module R (Mₚ r)]
      (f : (r : ↑s) → M →ₗ[R] Mₚ r) [inst_5 : ∀ (r : ↑s), IsLocalizedModule.Away (↑r) (f r)] {N P : Submodule R M},
      (∀ (r : ↑s),
          Submodule.localized₀ (Submonoid.powers ↑r) (f r) N ≤ Submodule.localized₀ (Submonoid.powers ↑r) (f r) P) →
        N ≤ P

Defined in

Mathlib.RingTheory.LocalProperties.Submodule

Cited by

2 results in Mathlib

Foundations

Depth 77 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringAddCommMonoidModuleAddCommMonoidModuleIsLocalizedModule.Away