How can I prove that every subring of $\mathbb{Q}$ is PID?
Asked
Active
Viewed 2,097 times
-1
-
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, so it may attract downvotes, or be put on hold. To prevent that, please [edit] the question. This 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
1 Answers
0
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.
Caleb Stanford
- 47,093