Finitely generated $R$-module as a sum of cyclic submodules

Question

I know that $M$ is finitely generated if there exist $a_1, ..., a_n$ in $M$ such that for any $x$ in $M$, there exist $r_1, ..., r_n$ in $R$ such that $x = r_1a_1 + \dots + r_na_n.$

But if I want to express this by its subsets is it correct to say, let $M$ be a finitely generated $R$-module, then $M = \sum_{i \in I}M_i$ where $I$ is a finite set of integers and $M_i \subseteq M$ for every $i$?

Answer 1

This would be correct if you require each $M_i$ to be the submodule of $M$ generated by a single element $a_i$ of $M$.