How can I prove that every subring of $\mathbb{Q}$ is PID?
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3What do you mean by PIK? Do you mean PID? – Loreno Heer Nov 09 '14 at 00:35

3Hello, welcome to Math.SE. Your question is phrased as an isolated problem, without any further information or context. This does not match [many users' quality standards](http://goo.gl/mLWc8), so it may attract downvotes, or be put on hold. To prevent that, please [edit] the question. [This](http://goo.gl/xQWVb) will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers. – Luiz Cordeiro Nov 09 '14 at 00:35

1An interesting question. *Is* every subring of $\Bbb Q$ a PID? At this point, doubtful I am. Still, no proof either way possess do I. – Robert Lewis Nov 09 '14 at 01:29
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Let $R$ be a subring of $\mathbb{Q}$. Then the function $f(a/b) := a$ is a Euclidean function on $R$. In particular, $R$ is a PID. (See also.)
A more general result has been linked, but deduction from Euclidean to PID is probably much easier in this case.