Theorems · Theorem · commutative algebra
Submodule.mem_torsionBySet_iff
∀ {R : Type u_1} {M : Type u_2} [inst : CommSemiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (s : Set R)
(x : M), x ∈ Submodule.torsionBySet R M s ↔ ∀ (a : ↑s), ↑a • x = 0- Defined in
- Mathlib.Algebra.Module.Torsion.Basic
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 28 from the axioms · uses propext, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- Modulestatement and proof · cited by 20,661
- AddCommMonoidstatement and proof · cited by 12,281
- CommSemiringstatement and proof · cited by 10,911
- Submodulestatement and proof · cited by 7,192
- Set.Elemstatement and proof · cited by 7,166
- Set.imageproof · cited by 5,609
- Set.mem_image_of_memproof · cited by 371
- Submodule.torsionBySetstatement and proof · cited by 34
- Submodule.torsionByproof · cited by 31
- Submodule.mem_sInfproof · cited by 11
Cited by5
Results whose statement or proof uses this declaration.
- Submodule.iSup_torsionBySet_ideal_eq_torsionBySet_iInfproof · cited by 5
- Module.isTorsionBySet_iff_torsionBySet_eq_topproof · cited by 3
- Submodule.torsionBySet_eq_torsionBySet_spanproof · cited by 2
- Submodule.torsion_gcproof · cited by 1
- Submodule.torsionBySet_isTorsionBySetproof · cited by 1