Nakayama's Lemma/Corollary 2

Corollary to Nakayama's Lemma

Let $A$ be a commutative ring with unity.

Let $M$ be a finitely generated $A$-module.

Let:

$m_1 + \operatorname{Jac} \left({A}\right) M, \dotsc, m_n + \operatorname{Jac} \left({A}\right) M$

generate $M / \operatorname{Jac} \left({A}\right) M$ over $A / \operatorname{Jac} \left({A}\right)$.

Then $m_1,\ldots, m_n$ generate $M$ over $A$.

Proof

Let $N$ be the submodule of $M$ generated by $m_1, \dotsc, m_n$.

Then:

$M = N + \operatorname{Jac} \left({A}\right) M$

Hence by Corollary 1:

$M = N$

$\blacksquare$