Theorems · Theorem · commutative algebra
Submodule.mem_annihilator
∀ {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {N : Submodule R M}
{r : R}, r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = 0- Defined in
- Mathlib.RingTheory.Ideal.Maps
- Cited by
- 10 results in Mathlib
- Foundations
- Depth 31 from the axioms · uses propext, Quot.sound
- Assumes
- SemiringAddCommMonoidModule
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
- Semiringstatement and proof · cited by 13,802
- AddCommMonoidstatement and proof · cited by 12,281
- Submodulestatement and proof · cited by 7,192
- Idealstatement · cited by 4,748
- Submodule.annihilatorstatement · cited by 42
Cited by10
Results whose statement or proof uses this declaration.
- Submodule.annihilator_iSupproof · cited by 2
- Submodule.annihilator_monoproof · cited by 2
- Submodule.annihilator_smulproof · cited by 2
- Submodule.mem_annihilator_spanproof · cited by 2
- Module.isTorsionBySet_annihilator_topproof · cited by 1
- Submodule.torsion_gcproof · cited by 1
- Ideal.bot_lt_annihilator_of_disjoint_nonZeroDivisorsproof · cited by 1
- Submodule.annihilator_map_mkQ_eq_colonproof · cited by 1
- Submodule.annihilator_botproof · cited by 0
- Submodule.mem_annihilator'proof · cited by 0