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I'm reading Matsumura's Commutative Ring Theory Thm7.10, which states

Let $(A,m,k)$ be a local ring, $M$ be a flat module over $A$ and $x_1,\cdots,x_n\in M$. If the images $\overline{x_1},\cdots,\overline{x_n}\in M/mM$ are linear independent over $k$, then $x_1,\cdots,x_n\in M$ are linear independent over $A$.

It also says

If $M$ is finitely generated or $m$ is nilpotent, then all minimal generating set of $M$ is a basis of $M$, and $M$ is free.

I proved the first statement and the image of minimal generating set goes to a basis of $M/mM$ over $k$ (with the hypothesis $A$ is a local ring). So the only thing left to be proved is that $M$ has a minimal generating set when $m$ is nilpotent. I'm stuck with this.

From this answer, it seems to be true. However I can't prove this. Can somebody help me?

I also want to know the example of a module which has no minimal generating set. Sorry for my poor English.

nessy
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  • A minimal generating set of $M$ means a subset $S\subseteq M$, which generates $M$ as $A$-module, and every subset $S'\subseteq S$ generates $M$ iff $S'=S$. – nessy Sep 04 '20 at 13:19
  • https://mathoverflow.net/questions/33513/non-finite-version-of-nakayamas-lemma Thanks to this page, I completed the proof of Thm7.10 (with an approach which is different from my first approach). I still wonder if my approach above can be completed. – nessy Sep 04 '20 at 14:40
  • https://math.stackexchange.com/questions/2023835 – user26857 Sep 04 '20 at 21:28

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