Theorems · Theorem · commutative algebra
Submodule.top_ne_ideal_smul_of_le_jacobson_annihilator
∀ {R : Type u_1} {M : Type u_2} [inst : CommRing R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] [Nontrivial M]
[Module.Finite R M] {I : Ideal R}, I ≤ (Module.annihilator R M).jacobson → ⊤ ≠ I • ⊤- Defined in
- Mathlib.RingTheory.Nakayama
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 80 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites14
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
- CommRingstatement and proof · cited by 17,173
- AddCommGroupstatement and proof · cited by 12,871
- Top.topstatement and proof · cited by 9,680
- Submodulestatement · cited by 7,192
- Idealstatement and proof · cited by 4,748
- Nontrivialstatement and proof · cited by 2,416
- Module.Finitestatement and proof · cited by 1,032
- Ideal.jacobsonstatement and proof · cited by 88
- Module.annihilatorstatement and proof · cited by 61
- Module.Finite.fg_topproof · cited by 28
- top_ne_botproof · cited by 15
Cited by6
Results whose statement or proof uses this declaration.
- IsSMulRegular.subsingleton_linearMap_iffproof · cited by 2
- Module.support_quotientproof · cited by 1
- Submodule.top_ne_set_smul_of_subset_jacobson_annihilatorproof · cited by 1
- RingTheory.Sequence.IsRegular.of_isWeaklyRegular_of_mem_maximalIdealproof · cited by 0
- RingTheory.Sequence.IsRegular.of_perm_of_subset_jacobson_annihilatorproof · cited by 0