# mathlib3documentation

ring_theory.nakayama

# Nakayama's lemma #

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This file contains some alternative statements of Nakayama's Lemma as found in Stacks: Nakayama's Lemma.

## Main statements #

• submodule.eq_smul_of_le_smul_of_le_jacobson - A version of (2) in Stacks: Nakayama's Lemma., generalising to the Jacobson of any ideal.

• submodule.eq_bot_of_le_smul_of_le_jacobson_bot - Statement (2) in Stacks: Nakayama's Lemma.

• submodule.smul_sup_eq_smul_sup_of_le_smul_of_le_jacobson - A version of (4) in Stacks: Nakayama's Lemma., generalising to the Jacobson of any ideal.

• submodule.smul_sup_eq_of_le_smul_of_le_jacobson_bot - Statement (4) in Stacks: Nakayama's Lemma.

Note that a version of Statement (1) in Stacks: Nakayama's Lemma can be found in ring_theory/noetherian under the name submodule.exists_sub_one_mem_and_smul_eq_zero_of_fg_of_le_smul

## Tags #

Nakayama, Jacobson

theorem submodule.eq_smul_of_le_smul_of_le_jacobson {R : Type u_1} {M : Type u_2} [comm_ring R] [ M] {I J : ideal R} {N : M} (hN : N.fg) (hIN : N I N) (hIjac : I J.jacobson) :
N = J N

Nakayama's Lemma* - A slightly more general version of (2) in Stacks 00DV. See also eq_bot_of_le_smul_of_le_jacobson_bot for the special case when J = ⊥.

theorem submodule.eq_bot_of_le_smul_of_le_jacobson_bot {R : Type u_1} {M : Type u_2} [comm_ring R] [ M] (I : ideal R) (N : M) (hN : N.fg) (hIN : N I N) (hIjac : I .jacobson) :
N =

Nakayama's Lemma* - Statement (2) in Stacks 00DV. See also eq_smul_of_le_smul_of_le_jacobson for a generalisation to the jacobson of any ideal

theorem submodule.smul_sup_eq_smul_sup_of_le_smul_of_le_jacobson {R : Type u_1} {M : Type u_2} [comm_ring R] [ M] {I J : ideal R} {N N' : M} (hN' : N'.fg) (hIJ : I J.jacobson) (hNN : N N' N I N') :
N I N' = N J N'

Nakayama's Lemma* - A slightly more general version of (4) in Stacks 00DV. See also smul_sup_eq_of_le_smul_of_le_jacobson_bot for the special case when J = ⊥.

theorem submodule.smul_sup_le_of_le_smul_of_le_jacobson_bot {R : Type u_1} {M : Type u_2} [comm_ring R] [ M] {I : ideal R} {N N' : M} (hN' : N'.fg) (hIJ : I .jacobson) (hNN : N N' N I N') :
I N' N

Nakayama's Lemma* - Statement (4) in Stacks 00DV. See also smul_sup_eq_smul_sup_of_le_smul_of_le_jacobson for a generalisation to the jacobson of any ideal