$R$ regarded as $R$-module

Question

Let $R=\prod_{n\in\mathbb{N}}\mathbb{Z}$ and $M$ be $R$ regarded as $R$-module is usual way.

Then the submodule $N=\bigoplus_{n\in\mathbb{N}}\mathbb{Z}$ is not finitely generated

Hint: for every $n\in\mathbb{N}$, $e_n=(0,…,0,1,0,…)\in N$ (the $1$ is on the $n-$th position).

Please help me to prove this result. Thanks in advance

Answer 1

Another hint: An element of $N$ has only finitely many non-zero entries. If $N$ is finitely generated, then the collection of all linear combination of those generators can only make finitely many of the $e_n$.