# Proper $\Bbb Z$-submodules of $\Bbb Q$ are finitely generated or not? [closed]

Let $$M$$ be a proper $$\Bbb Z$$-submodule of $$\Bbb Q.$$ Can we say that $$M$$ is finitely generated?

I know that $$\Bbb Q$$ is not finitely generated as a $$\Bbb Z$$-module.

Please help me in this regard. Thank you very much.

## closed as off-topic by Najib Idrissi, Arnaud D., Shailesh, abc..., stressed outFeb 15 at 2:58

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• What about the fractions with odd denominators? – Lord Shark the Unknown Feb 14 at 7:25
• @Lord Shark the Unknown yeah it's a infinitely generated submodule. – Dbchatto67 Feb 14 at 7:34
• It can be finitely generated or not. In addition to other answers, consider $\mathbb{Z}$ as a $\mathbb{Z}$-submodule of $\mathbb{Q}$. Obviously, it's free with basis $\{1\}$. – stressed out Feb 14 at 7:50
• @Dbchatto67 what's your question now? It got answered twice and you just said "Yes, I know."? – Paul K Feb 14 at 7:55
• I have got my answer from the hint given by @Lord Shark the Unknown. – Dbchatto67 Feb 15 at 3:19