Theorems · Theorem · commutative algebra
Module.mem_annihilator
∀ {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {r : R},
r ∈ Module.annihilator R M ↔ ∀ (m : M), r • m = 0- Defined in
- Mathlib.RingTheory.Ideal.Maps
- Cited by
- 14 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.
Cites8
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Modulestatement and proof · cited by 20,661
- Semiringstatement and proof · cited by 13,802
- AddCommMonoidstatement and proof · cited by 12,281
- Idealstatement · cited by 4,748
- AddMonoidHom.extproof · cited by 149
- AddMonoid.Endproof · cited by 64
- Module.annihilatorstatement and proof · cited by 61
Cited by14
Results whose statement or proof uses this declaration.
- Module.End.isSemisimple_of_squarefree_aeval_eq_zeroproof · cited by 4
- LinearMap.annihilator_le_of_injectiveproof · cited by 3
- Polynomial.span_minpoly_eq_annihilatorproof · cited by 3
- LocalizedModule.exists_subsingleton_awayproof · cited by 2
- LinearMap.annihilator_le_of_surjectiveproof · cited by 2
- Module.annihilator_eq_botproof · cited by 2
- Module.annihilator_eq_top_iffproof · cited by 2
- IsSMulRegular.linearMap_subsingleton_of_mem_annihilatorproof · cited by 1
- IsSMulRegular.notMem_of_mem_minimalPrimesproof · cited by 1
- QuotSMulTop.mem_annihilatorproof · cited by 1
- Module.isTorsionBySet_annihilatorproof · cited by 0
- Module.isTorsionBy_iff_mem_annihilatorproof · cited by 0