- #1

steenis

- 312

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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.$

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.$

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