Theorems · Theorem · commutative algebra
Submodule.eq_bot_of_eq_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] {I : Ideal R}
{N : Submodule R M}, N.FG → N = I • N → I ≤ N.annihilator.jacobson → N = ⊥- Defined in
- Mathlib.RingTheory.Nakayama
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 79 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommRingAddCommGroupModule
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
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
- Submodulestatement and proof · cited by 7,192
- Idealstatement and proof · cited by 4,748
- Bot.botstatement · cited by 4,720
- Eq.leproof · cited by 605
- Submodule.FGstatement and proof · cited by 230
- Ideal.jacobsonstatement and proof · cited by 88
- Submodule.annihilatorstatement and proof · cited by 42
- Submodule.eq_smul_of_le_smul_of_le_jacobsonproof · cited by 3
- Submodule.annihilator_smulproof · cited by 2
Cited by4
Results whose statement or proof uses this declaration.
- Submodule.top_ne_ideal_smul_of_le_jacobson_annihilatorproof · cited by 6
- Submodule.eq_bot_of_eq_pointwise_smul_of_mem_jacobson_annihilatorproof · cited by 1
- IsLocalRing.maximalIdeal_sq_lt_maximalIdealproof · cited by 1
- Submodule.eq_bot_of_set_smul_eq_of_subset_jacobson_annihilatorproof · cited by 0