What the rings between Principal Ideal Ring and Field of fractions are?

Question

Prove that all of the rings, which mediate between principal ideal ring $K$ and the field of fractions $Q$, are the principal ideal ring.

Does "mediate between" mean that the ring, say $R$, is contained in $Q$ and contains $K$? I.e. $K \subseteq R \subseteq Q$? — user50948, Jun 07 '14 at 14:25
Related quasi duplicate of http://math.stackexchange.com/questions/536624/is-the-localization-of-a-pid-a-PID which might help also — rschwieb, Jun 07 '14 at 15:21

score 3 · Answer 1 · answered Jun 07 '14 at 14:53

3

Hint $ $ They're localizations since $\,K[a/b] = K[1/b],\,$ by $\,(a,b) =1\,\Rightarrow\, ra+sb = 1\,\Rightarrow\, ra/b + s = 1/b$

answered Jun 07 '14 at 14:53

Bill Dubuque

score 2 · Answer 2 · answered Jun 07 '14 at 14:56

2

$K = \mathbb{Z}$

$R = \{\frac{a}{2^n} : a \in \mathbb{Z}, n \in \mathbb{N}\}$

$Q = \mathbb{Q}$

If I understand your question here is an counter example.

answered Jun 07 '14 at 14:56

John Machacek

Isn't $R$ here also PID? – user50948 Jun 07 '14 at 15:03
Yes. I read you question as that there only rings between a principle ideal ring and it's field of fractions are THE principle ideal ring to mean there are no proper rings between. If you are asking if any in between must also be A principle ideal ring that is true. – John Machacek Jun 07 '14 at 15:09
I don't think this should have been downvoted. The question is poorly worded and this is also what I thought the OP was asking. – Seth Jun 07 '14 at 15:23

2 Answers2