About Nakayama Lemma

Question

I am trying to study Nakayama's Lemma but I am confused which version of Nakayama's Lemma should I study? There are so many versions of it and so many references that I am getting confused.

I would appreciate if anyone can advice which version and which reference to follow.

Answer 1

I am new to Commutative Algebra as well, I wanted to present my understanding of the NAK Lemma. The version I am familiar with is this from Atiyah- MacDonald:

Let $A$ be a commutative ring with identity. Let $M$ be an $A - $ module and consider an ideal $I \subseteq J(A)$ where $J(A)$ is the Jacobson's radical of $A$. Suppose that $IM=M $ then $M=0$.

To understand better, consider a local ring $(A,\mathbb{m})$. NAK lemma simply says $IM \neq M$ except when $M=0$. As to intuition, in a local ring, $J(A)=\mathbb{m}$. Then for any ideal $I \subseteq \mathbb{m}$, $1\notin I$ and $\mathbb{m}$ has no units. Therefore the module $IM$ can never generate $M$. $IM$ is analogous to taking span of a subset in a vector space.