Nakayama's Lemma/Corollary 1

Corollary to Nakayama's Lemma

Let $A$ be a commutative ring with unity.

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

Let there exist a submodule $N \subseteq M$ such that:

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

Then $M = N$.

Proof

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

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

so by Nakayama's Lemma:

$M/N = 0$

and so:

$M = N$

$\blacksquare$