Theorems · Definition · commutative algebra
Submodule.torsionBy
(R : Type u_1) → (M : Type u_2) → [inst : CommSemiring R] → [inst_1 : AddCommMonoid M] → [inst_2 : Module R M] → R → Submodule R M
The a-torsion submodule for a in R, containing all elements x of M such that
a • x = 0.
- Defined in
- Mathlib.Algebra.Module.Torsion.Basic
- Cited by
- 31 results in Mathlib
- Foundations
- Depth 26 from the axioms · uses propext, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Modulestatement and proof · cited by 20,661
- AddCommMonoidstatement and proof · cited by 12,281
- CommSemiringstatement and proof · cited by 10,911
- Submodulestatement · cited by 7,192
- LinearMap.kerproof · cited by 848
- DistribSMul.toLinearMapproof · cited by 50
Cited by34
Results whose statement or proof uses this declaration.
- Submodule.torsionBySetproof · cited by 34
- Submodule.torsionBySet_span_singleton_eqstatement · cited by 6
- Submodule.torsionBySet_le_torsionBySet_of_subsetproof · cited by 5
- Submodule.torsionBySet_singleton_eqstatement · cited by 5
- Submodule.mem_torsionBySet_iffproof · cited by 5
- AddSubgroup.torsionByproof · cited by 5
- Submodule.mem_torsionBy_iffstatement · cited by 4
- Submodule.smul_torsionBystatement and proof · cited by 4
- Module.equiv_directSum_of_isTorsionproof · cited by 2
- Module.isTorsionBy_iff_torsionBy_eq_topstatement · cited by 2
- Submodule.isInternal_prime_power_torsion_of_pidstatement and proof · cited by 1