coe_torsion_eq_annihilator_ne_bot

English

NoZeroSMulDivisors on quotient by torsion holds in the quotient by torsion.

Русский

NoZeroSMulDivisors на квотентной модульной структуре сохраняется в фактор-модулеtorsion.

LaTeX

$$NoZeroSMulDivisors R (M ⧸ torsion R M)$$

Lean4

theorem coe_torsion_eq_annihilator_ne_bot : (torsion R M : Set M) = {x : M | (R ∙ x).annihilator ≠ ⊥} :=
  by
  ext x; simp_rw [Submodule.ne_bot_iff, mem_annihilator, mem_span_singleton]
  exact
    ⟨fun ⟨a, hax⟩ =>
      ⟨a, fun _ ⟨b, hb⟩ => by rw [← hb, smul_comm, ← Submonoid.smul_def, hax, smul_zero],
        nonZeroDivisors.coe_ne_zero _⟩,
      fun ⟨a, hax, ha⟩ => ⟨⟨_, mem_nonZeroDivisors_of_ne_zero ha⟩, hax x ⟨1, one_smul _ _⟩⟩⟩

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