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February 7th, 2018, 01:52 PM   #1
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Finitely generated submodule

Please help me to prove the following resut:

Let $R=\prod_{n\in\mathbb{N}}\mathbb{Z}$ and $M$ be $R$ regarded as $R$-module is usual way.

Then the submodule $N=\bigoplus_{n\in\mathbb{N}}\mathbb{Z}$ is not finitely generated

Hint: for every $n\in\mathbb{N}$, $e_n=(0,…,0,1,0,…)\in N$ (the $1$ is on the $n-$th position)


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Last edited by mona123; February 7th, 2018 at 02:07 PM.
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February 10th, 2018, 07:00 PM   #2
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If f is any element of N, there is a natural number n such that if $k\geq n$ then $f_k=0$ -- defining property of a direct sum. So if S is any finite subset of N, there is m such that for any $f\in S$ and $k\geq m$, $f_k=0$. It should now be obvious.
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