Theorems · Theorem · commutative algebra

eq_bot_of_generator_maximal_submoduleImage_eq_zero

∀ {R : Type u} {M : Type v} [inst : Ring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] {ι : Type u_1}
  {N O : Submodule R M} (b : Module.Basis ι R ↥O),
  N ≤ O →
    ∀ {ϕ : ↥O →ₗ[R] R},
      (∀ (ψ : ↥O →ₗ[R] R), ¬ϕ.submoduleImage N < ψ.submoduleImage N) →
        ∀ [inst_3 : (ϕ.submoduleImage N).IsPrincipal], Submodule.IsPrincipal.generator (ϕ.submoduleImage N) = 0 → N = ⊥

Defined in

Mathlib.LinearAlgebra.FreeModule.PID

Cited by

1 results in Mathlib

Foundations

Depth 76 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingAddCommGroupModuleSubmodule.IsPrincipal