Theorems · Theorem · commutative algebra

Submodule.smul_le_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 N' : Submodule R M}, N'.FG → I ≤ ⊥.jacobson → N' ≤ N ⊔ I • N' → I • N' ≤ N

Nakayama's Lemma - Statement (4) in [Stacks 00DV](https://stacks.math.columbia.edu/tag/00DV). See also sup_smul_eq_sup_smul_of_le_smul_of_le_jacobson for a generalisation to the jacobson of any ideal

Defined in

Mathlib.RingTheory.Nakayama

Cited by

0 results in Mathlib

Foundations

Depth 89 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModule