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Give an example to show that an increasing union of finitely generated submodules of some module need not be finitely generated.

My attempt:

Let $R = \Bbb Z$ .

Consider $M=\Bbb Z \oplus \Bbb Z(\sqrt{2})+\oplus \Bbb Z(\sqrt{3})\oplus \ldots$ and the chain of submodules :$$\Bbb Z \subset \Bbb Z \oplus \Bbb Z(\sqrt{2}) \subset \Bbb Z \oplus \Bbb Z(\sqrt{2}) \oplus \Bbb Z(\sqrt{3}) \subset \dots $$

Each sub-module in the chain is finitely generated but their union (which turns out to be $M$ ) is not!

Please point out mistakes if there are any in my argument.

I have observed that there are a few questions already in the same query but the main reason for posting this question is to see whether my argument is valid or not. So please do not mark it as duplicate.

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  • $\begingroup$ For extra fun, why not make all the f.g. submodules cyclic (that is with one generator). $\endgroup$ – Lord Shark the Unknown Oct 3 '18 at 17:36
  • $\begingroup$ @LordSharktheUnknown Okay I'll try doing it. But is my answer alright? $\endgroup$ – Utsav Dewan Oct 3 '18 at 17:40
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Your answer is correct, though it is more complicated than it needs to be. You could just let your chain be $\mathbb{Z} \subseteq \mathbb{Z}^2 \subseteq \mathbb{Z}^3 \subseteq \cdots$, where $\mathbb{Z}^n$ is the direct sum of $n$ copies of $\mathbb{Z}$. (To be really precise here, you should specify how $\mathbb{Z}^n$ is a submodule of $\mathbb{Z}^{n+1}$.)

Or start with, say the polynomial ring $\mathbb{Z}[X]$, viewed as a $\mathbb{Z}$-module, and find an increasing sequence of finitely-generated submodules whose union is $\mathbb{Z}$.

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