English
In a domain, NoZeroSMulDivisors is equivalent to torsion being bot after quotient.
Русский
В области (domain) NoZeroSMulDivisors эквивалентно тому, что при квартиовании тorsion становится ⊥.
LaTeX
$$NoZeroSMulDivisors R M ↔ torsion R M = ⊥$$
Lean4
theorem _root_.Submodule.annihilator_top_inter_nonZeroDivisors [Module.Finite R M] (hM : Module.IsTorsion R M) :
((⊤ : Submodule R M).annihilator : Set R) ∩ R⁰ ≠ ∅ :=
by
obtain ⟨S, hS⟩ := ‹Module.Finite R M›.fg_top
refine Set.Nonempty.ne_empty ⟨_, ?_, (∏ x ∈ S, (@hM x).choose : R⁰).prop⟩
rw [Submonoid.coe_finset_prod, SetLike.mem_coe, ← hS, mem_annihilator_span]
intro n
letI := Classical.decEq M
rw [← Finset.prod_erase_mul _ _ n.prop, mul_smul, ← Submonoid.smul_def, (@hM n).choose_spec, smul_zero]