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.

Answer 1

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

Answer 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.