Theorems · Definition · commutative algebra
Submodule.annihilator
{R : Type u_1} →
{M : Type u_2} → [inst : Semiring R] → [inst_1 : AddCommMonoid M] → [inst_2 : Module R M] → Submodule R M → Ideal RN.annihilator is the ideal of all elements r : R such that r • N = 0.
- Defined in
- Mathlib.RingTheory.Ideal.Maps
- Cited by
- 42 results in Mathlib
- Foundations
- Depth 29 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
- Module.annihilatorproof · cited by 61
Cited by42
Results whose statement or proof uses this declaration.
- Submodule.mem_annihilatorstatement · cited by 10
- Submodule.annihilator_topstatement · cited by 7
- LocalizedModule.subsingleton_iff_support_subsetproof · cited by 5
- IsSemisimpleModule.annihilator_isRadicalproof · cited by 4
- Module.mem_support_iff_exists_annihilatorstatement and proof · cited by 4
- Module.mem_support_iff_of_finiteproof · cited by 4
- Submodule.eq_bot_of_eq_ideal_smul_of_le_jacobson_annihilatorstatement and proof · cited by 4
- Submodule.isInternal_prime_power_torsionstatement and proof · cited by 2
- Submodule.mem_annihilator_spanstatement · cited by 2
- Algebra.FormallyUnramified.iff_exists_tensorProductproof · cited by 2
- Submodule.annihilator_eq_top_iffstatement · cited by 2
- Submodule.annihilator_iSupstatement and proof · cited by 2