On a finiteley generated submodule of a direct sum of left R-modules

[steenis]

Suppose $(M_i)_{i \in I}$ is a family of left $R$-modules and $M = \bigoplus_{i \in I} M_i$.
Suppose $N = \langle x_1 \cdots x_m \rangle$ is a finitely generated submodule of $M$.
Then for each $j = 1 \cdots m$, there is a finite $I_j \subset I$ such that $x_j \in \bigoplus_{i \in I_j} M_i$.

Can anyone help me with this ?*EDIT*
It is clear that each $x_j$ is in at least one $M_i$. I even think that it is exactly one. I need some confirmation.$

 

[steenis]

My *EDIT* in post #1 in not correct.It took me a while, but the solution is:

Each $x_j$ is an element of $M = \bigoplus_{ i \in I} M_i $.

Therefore $x_j = (m_{ij})_{I \in I}$ where $m_{ij} \in M_i$ is the i-th component of $x_j$ in $\bigoplus_{ i \in I} M_i $.

M is an external direct sum, so only finitely many $m_{ij}$ are nonzero.

Let $I_j = \{i \in I | m_{ij} \neq 0 \}$, $I_j$ is finite.

Then $x_j \in $ $\bigoplus_{ i \in I_j} M_i $.