A question from Hungerford’s algebra, Chapter VIII, lemma 4.2

Question

I am reading Hungerford’s algebra and I have a question about the lemma 4.2.

The lemma 4.2:

Let B be a finitely generated module over a commutative ring $R$ with identity and let $I=\left\{r\in R|rb=0, \forall b \in B\right\}$ be the annihilator of $B$ in $R$. Then B satisfies the ascending chain condition on submodules if and only if $R/I$ is a Noetherian ring.

The following is the author’s sketch of the proof.

The first part of the proof uses $B$ is a finitely generated module.

I don't think the second part of the proof uses $B$ is a finitely generated module.

Meanwhile, in the first part of the prooof, $B$ satisfies the ascending chain condition , then $B$ is already a Noetherian module.

So, Why does the author assume $B$ is a finitely generated module and I would like to know if it can be removed?

Thanks!

Answer 1

Check the statement of Theorem 1.8. It says

Over a noetherian ring with identity, every finitely generated unitary module satisfies ACC.

So, $B$ being finitely generated is used in the last sentence of the proof of $\Leftarrow$.