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This was a statement on wikipedia's page on finitely generated module:

Any module is a union of an increasing chain of finitely generated submodules.

But if we look at $\mathbb{R}$ as a $\mathbb{Q}$-module, a chain contains only countably many submodules, and each submodule only contains countably many elements - how can their union by $\mathbb{R}$? Am I getting some definitions wrong?

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    $\begingroup$ Following the suggestion of @JeremyRickard in his reply (+1), I have changed the sentence in the Wikipedia page to Any module is the union of the directed set of its finitely generated submodules. $\endgroup$ Feb 4, 2017 at 13:54
  • $\begingroup$ @AndreasCaranti Thanks for doing that. One day I'll learn how to edit Wikipedia ... $\endgroup$ Feb 4, 2017 at 13:57
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    $\begingroup$ @JeremyRickard, you're welcome, and thanks for pointing out where the problem was. Editing on the Wikipedia is absolutely straightforward, if you can edit here, you can edit there. I find Maths on the English version of the Wikipedia to be of excellent quality - after all, people like Terence Tao and Tim Gowers use it as a standard reference. But whenever one still finds a mistake, it is easy to correct it. $\endgroup$ Feb 4, 2017 at 14:03

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Wikipedia is wrong here. I suspect that they've just misstated the fact (which is a fact) that every module is the union of a filtered set of finitely generated submodules.

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  • $\begingroup$ Oops yes, I misdiagnosed the mistake that was occurring. It was actually this. $\endgroup$
    – rschwieb
    Feb 4, 2017 at 13:55

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