Let $R$ be a ring, $M$ be a nontrivial proper left $R$-module and $(M_\lambda)_{\lambda \in \Lambda}$ be a family of submodules of $M$ such that $M = \sum_{\lambda \in \Lambda} M_\lambda$.
If $M$ is finitely generated, show that $M$ is a finite sum of those $M_\lambda$.
I really don't know what to do, this seems intuitive, but I have no clue on how to proceed.
Any help would be welcome.
This one is easy. Suppose that a set of generators for $M$ is given by $m_1, \ldots, m_n \in M$. For each $j \in \{1, \ldots, n\}$, write $$ m_j = r_{j, \lambda_1} x_{j, \lambda_1} + \cdots + r_{j, \lambda_{k_j}} x_{j, \lambda_{k_j}} $$ with $x_{j, \lambda_l} \in M_{\lambda_l}$. Then the elements $x_{j, \lambda_l}$ obviously generate all of $M$, and there are finitely many of them, so that there are finitely many $M_{\lambda_l}$'s which are necessary to generate $M$.