Let $A$ be a local ring with nilpotent maximal ideal $\mathfrak{m}$. \mathfrak{m}$ (i.e., some power of $\mathfrak{m}$ vanishes), and $M$ an $A$-module (not necessarily finitely generated). Let $\bar{S}\subset M/\mathfrak{m}M$ be a set of generators and $S$ its preimage a set of representatives of $\bar{S}$ in M. $M$. Then is it true that S $S$ is again a set of generators in of $M$? This is a common form of Nakayama's lemma with the assumption of finite generation of $M$ replaced by replacing the nilpotence of $\mathfrak{m}$. A passage in Matsumura's book "Commutative Ring Theory" (see Theorem 7.10) seems to be implying imply this result, and I can't figure out why.
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Non-finite version of Nakayama's lemma?$A$ a local ring with nilpotent maximal ideal $\mathfrak{m}$. $M$ an $A$-module (not necessarily finitely generated). Let $\bar{S}\subset M/\mathfrak{m}M$ be a set of generators and $S$ its preimage in M. Then is it true that S is again a set of generators in $M$? This is a common form of Nakayama's lemma with the assumption of finite generation of $M$ replaced by nilpotence of $\mathfrak{m}$. A passage in Matsumura's book seems to be implying this result, and I can't figure out why.
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