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 February 7th, 2018, 01:52 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 02:07 PM. February 10th, 2018, 07:00 PM #2 Member   Joined: Jan 2016 From: Athens, OH Posts: 93 Thanks: 48 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. Tags finitely, finitly, generated, submodule Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post Ezio Abstract Algebra 0 September 4th, 2015 03:06 AM martexel Topology 1 September 4th, 2010 12:48 PM kennedy Linear Algebra 1 October 1st, 2009 04:59 PM payman_pm Abstract Algebra 7 September 19th, 2007 05:21 AM sastra81 Abstract Algebra 0 January 3rd, 2007 07:54 AM

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