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February 7th, 2018, 12:52 PM  #1 
Senior Member Joined: Jan 2015 From: usa Posts: 104 Thanks: 1  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) Thanks Last edited by mona123; February 7th, 2018 at 01:07 PM. 
February 10th, 2018, 06:00 PM  #2 
Member Joined: Jan 2016 From: Athens, OH Posts: 92 Thanks: 47 
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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finitely, finitly, generated, submodule 
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