Theorems · Definition · commutative algebra
Submodule.torsion
(R : Type u_1) → (M : Type u_2) → [inst : CommSemiring R] → [inst_1 : AddCommMonoid M] → [inst_2 : Module R M] → Submodule R M
The torsion submodule, containing all elements x of M such that a • x = 0 for some
non-zero-divisor a in R.
- Defined in
- Mathlib.Algebra.Module.Torsion.Basic
- Cited by
- 15 results in Mathlib
- Foundations
- Depth 19 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
- nonZeroDivisorsproof · cited by 895
- Submodule.torsion'proof · cited by 6
Cited by15
Results whose statement or proof uses this declaration.
- Module.Flat.torsion_eq_botstatement and proof · cited by 2
- Module.equiv_free_prod_directSumproof · cited by 2
- finrank_quotient_eq_of_le_torsionstatement and proof · cited by 1
- finrank_quotient_torsion_eqproof · cited by 1
- Submodule.torsion_intstatement and proof · cited by 1
- Submodule.torsion_isTorsionstatement · cited by 1
- rank_quotient_eq_of_le_torsionstatement and proof · cited by 1
- Module.Flat.flat_iff_torsion_eq_bot_of_isBezoutstatement and proof · cited by 1
- Module.Flat.flat_iff_torsion_eq_bot_of_valuationRing_localization_isMaximalstatement and proof · cited by 1
- Submodule.isTorsionFree_iff_torsion_eq_botstatement · cited by 1
- Submodule.coe_torsion_eq_annihilator_ne_botstatement and proof · cited by 0
- IsDedekindDomain.flat_iff_torsion_eq_botstatement · cited by 0