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I'm a bit stuck here. I have a problem tha goes like this:

Let R be a principal ideal domain (PID). Let D a subset of R

which is multiplicatively closed. Show that the ring of quotients D^(-1)R is

a PID too.

I've tried several different ways but I couldn't get to the result.

For example, take an ideal I of D^(-1)R. I was able to show that

all r \in R such as r/d \in I for some d \in D, form an ideal of R,

(Thus a principal ideal). If I could show that I=(a)/d for some d..

Anyway, I don't know..

Any hints for that?

Thank you in advance!