$A$ 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.