TITLE: Favourite applications of the Nakayama Lemma
QUESTION [20 upvotes]: Inspired by a recent question on the nilradical of an absolutely flat ring, what are some of your favourite applications of the Nakayama Lemma? It would be good if you outlined a proof for the result too. I am also interested to see the Nakayama Lemma prove some facts in Algebraic Geometry if possible. Here are some facts which one can use the Nakayama lemma to prove.

A local ring that is absolutely flat is a field - proof given here.
Every set of $n$  - generators for a free module of rank $n$ is a basis - proof given here.
For any integral domain $R$ (that is not a field) with fraction field $F$, it is never the case that $F$ is a f.g. $R$ - module. Sketch proof: if $F$ is f.g. as a $R$ - module then certainly it is f.g. as a $R_{\mathfrak{m}}$ module for any maximal ideal $\mathfrak{m}$. Then $\mathfrak{m}_{\mathfrak{m}}F = F$ and so Nakayama's Lemma implies $F = 0$ which is ridiculous.

REPLY [2 votes]: Let $A$ be a noetherian ring. $\mathfrak{a} \subset A$ an ideal and $M$ be a finitely generated $A$-Module.
Let $N = \bigcap_{n \in \mathbb{N}} \mathfrak{a}^nM$.
Then there exists $x \in A$ such that $x \equiv 1 \mod\mathfrak{a}$ with $x \cdot N = (0)$.
In particular, if $x \in rad(A)$, then it is $N=(0)$.
Corollar Let $A$ be noetherian integral domain. $\mathfrak{a} \subset A$ an ideal with $\mathfrak{a} \neq A$. Then it holds
$\bigcap_{n \in \mathbb{N}} \mathfrak{a}^n = (0)$
(proof by Artin-Rees Lemma and Nakayama Lemma)