Exercise 111.8.1. Let $A$ be a ring. Let $I$ be an ideal of $A$. Let $M$ be an $A$-module. Let $x_1, \ldots , x_ n \in M$. Assume that

$M/IM$ is generated by $x_1, \ldots , x_ n$,

$M$ is a finite $A$-module,

$I$ is contained in every maximal ideal of $A$.

Show that $x_1, \ldots , x_ n$ generate $M$. (Suggested solution: Reduce to a localization at a maximal ideal of $A$ using Exercise 111.7.2 and exactness of localization. Then reduce to the statement of Nakayama's lemma in the lectures by looking at the quotient of $M$ by the submodule generated by $x_1, \ldots , x_ n$.)

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