Theorems · Theorem · commutative algebra
Submodule.eq_bot_of_le_smul_of_le_jacobson_bot
∀ {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 ≤ ⊥.jacobson → N = ⊥Nakayama's Lemma - Statement (2) in
[Stacks 00DV](https://stacks.math.columbia.edu/tag/00DV).
See also eq_smul_of_le_smul_of_le_jacobson for a generalisation
to the jacobson of any ideal
- Defined in
- Mathlib.RingTheory.Nakayama
- Cited by
- 3 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.
Cites10
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 and proof · cited by 4,720
- Submodule.FGstatement and proof · cited by 230
- Ideal.jacobsonstatement and proof · cited by 88
- Submodule.bot_smulproof · cited by 11
- Submodule.eq_smul_of_le_smul_of_le_jacobsonproof · cited by 3
Cited by3
Results whose statement or proof uses this declaration.
- Ideal.ramificationIdx'_eq_one_of_map_localizationproof · cited by 3
- Ideal.height_le_one_of_isPrincipal_of_mem_minimalPrimes_of_isLocalRingproof · cited by 1
- Module.free_quotSMulTop_iff_freeproof · cited by 0