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I am new on this forum, this is my gift for you.

Suppose ##(M_i)_{i \in I}## is a family of left ##R##-modules and ##M = \bigoplus_{i \in I} M_i## (external direct sum).

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 the proof of this?

This is from a book of "Ribenboim – Rings and modules (1969)". This is part of the proof of (d) on p.21 (chapter I, section 6).

Suppose ##(M_i)_{i \in I}## is a family of left ##R##-modules and ##M = \bigoplus_{i \in I} M_i## (external direct sum).

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 the proof of this?

This is from a book of "Ribenboim – Rings and modules (1969)". This is part of the proof of (d) on p.21 (chapter I, section 6).

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