Question about a submodule

Question

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

The submodule $N=\bigoplus_{n\in\mathbb{N}}\mathbb{Z}$ is it finitely generated?

Thanks

Answer 1

An element of $N$ is only nonzero in finitely many coordinates.

A finite set of elements of $N$, therefore, can only generate a submodule that is nonzero only on finitely many coordinates. (All the coordinates where the generators are nonzero.)

But $N$ is nonzero in infinitely many coordinates.