Question about submodule

Question

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

Why $M$ is cyclic?

Thanks

Answer 1

Any (unital) ring $R$ considered as an $R$-module is always cyclic. It is generated by the single generator $1$: Every element $r\in R$ can be written as $r⋅1$