---
tag: 15JHBYR6KH
kind: statement
role: lemma
name: Nakayama's lemma
msc: [13C05, 13H99]
uses: [0310REJJY8, 0NZ8JA25WK, 05QY1VEBYD, XA0DCNH7WY, 096Z9MP2ME, MZ02BKPHJA]
about: [05QY1VEBYD, XA0DCNH7WY]
sources:
  - url: https://en.wikipedia.org/wiki/Nakayama%27s_lemma
same_as:
  wikidata: Q1399751
  wikipedia: Nakayama's lemma
---

## Statement

Let $R$ be a [[commutative ring|0310REJJY8]] with [[Jacobson radical|XA0DCNH7WY]] $J(R)$.
Let $I \subseteq J(R)$ be an [[ideal|096Z9MP2ME]] of $R$, and let $M$ be a [[finitely generated|05QY1VEBYD]] $R$-[[module|0NZ8JA25WK]].
If $IM = M$, then $M = 0$.

## Remarks

Proof idea: let $m_1, \ldots, m_n$ generate $M$ with $n$ minimal, and suppose $n \ge 1$.
From $IM = M$, write $m_n = a_1 m_1 + \cdots + a_n m_n$ with $a_i \in I$.
Then $(1 - a_n) m_n$ lies in the submodule generated by $m_1, \ldots, m_{n-1}$.
Since $a_n \in J(R)$, the element $1 - a_n$ is a unit, so $m_n$ is not needed, contradicting minimality.
Another proof uses the [[determinant trick|A8HRT9H3XA]] to find $a \in I$ with $(1 + a) M = 0$.

Consequence: if $N \subseteq M$ is a [[submodule|05WDSN597C]] and $M = N + IM$, then $M = N$; apply the lemma to $M/N$.
For a [[local ring|MZ02BKPHJA]] $(R, \mathfrak m)$ with $I = \mathfrak m$, this says that elements of $M$ generate $M$ exactly when their images span the vector space $M/\mathfrak m M$ over $R/\mathfrak m$.

Both hypotheses are needed.
Without finite generation: over $R = \Z_{(p)}$ with $I = p\Z_{(p)} = J(R)$, the module $M = \Q$ satisfies $pM = M$ but $M \ne 0$.
Without $I \subseteq J(R)$: over $R = \Z$ with $I = 2\Z$, the module $M = \Z/3\Z$ satisfies $2M = M$, since $2$ is invertible modulo $3$, but $M \ne 0$.

## Explanations

### L1 human

Over a local ring, the maximal ideal consists of "small" elements.
Nakayama's lemma says that a finitely generated module cannot be made of small multiples of itself unless it is zero.
In practice it says that to find generators of a module, it is enough to find generators "modulo the maximal ideal", which is a question of linear algebra.

### L2 human

The lemma is the main tool for passing from the [[residue field|NJ3APQ8BHH]] back to the local ring.
It gives the equality between the minimal number of generators of a finitely generated module $M$ over a local ring and $\dim_k M/\mathfrak m M$.
With $M = \mathfrak m$, this identifies the [[embedding dimension|DP3KSCRKFP]] with the minimal number of generators of $\mathfrak m$.
It also shows that [[finitely generated projective modules over local rings are free|942FK83S09]], and it enters the proof of the [[Krull intersection theorem|1Q1AH2V41S]].