Nakayama's lemma

Main results

• exists_sub_one_mem_and_smul_eq_zero_of_fg_of_le_smul is Nakayama's lemma, in the following form: if N is a finitely generated submodule of an ambient R-module M and I is an ideal of R such that N ⊆ IN, then there exists r ∈ 1 + I such that rN = 0.

theorem Submodule.exists_sub_one_mem_and_smul_eq_zero_of_fg_of_le_smul {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (I : Ideal R) (N : Submodule R M) (hn : N.FG) (hin : N ≤ I • N) : ∃ (r : R), r - 1 ∈ I ∧ ∀ n ∈ N, r • n = 0

Nakayama's Lemma. Atiyah-Macdonald 2.5, Eisenbud 4.7, Matsumura 2.2.

Stacks Tag 00DV

theorem Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (I : Ideal R) (N : Submodule R M) (hn : N.FG) (hin : N ≤ I • N) : ∃ r ∈ I, ∀ n ∈ N, r • n = n