If we have an integral domain R such that Z (integers) $\displaystyle \subset$ R $\displaystyle \subset$ Q (rationals), how would I go about showing that R is a principal ideal domain? I need to show that for any ideal I in R, then I = aR for some a in R... but how do you show this with a ring that include all the integers, and a few rationals thrown in?

I thought that if I could show that R was a Euclidean domain, then it would follow that R was a PID. But, while I could find d functions defined on Z (d(x) = |x|) and Q (d(x) = 1), neither of these would work properly for R.