Showing a localization is a principal ideal domain (non-trivial problem)

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Homework Statement



Let [itex]R[/itex] be an integral domain. Say a prime [itex]p \in R[/itex] is small if [itex]a\in\bigcap\limits _{n=1}^{\infty}\left\langle p^{n}\right\rangle = \left\langle 0\right\rangle[/itex]

Show that if [itex]p[/itex] is a small prime and [itex]D = R \setminus \left\langle p\right\rangle[/itex] then [itex]R_D[/itex] is a principal ideal domain.

Homework Equations



Some basic facts... ED implies PID, field implies PID, PID implies UFD. Localization is a PID if R is a PID, etc.

The Attempt at a Solution



I tried using First Isomorphism Theorem to show [itex]R_D[/itex] is a field; I quickly shot this down with a counterexample by taking R to be the integers, and since every prime is small in the integers, taking p = 2 shows that 2/3 in the localization has no inverse.

The only other approach I can think of that can be used when I know squat about the ring is to show it is an ED. But I can't find a valid Euclidean function.

Any ideas? I don't want solutions, obviously... just a tiny hint to push me in a promising direction.
 
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on Phys.org
Thank you for the hint! Will see where it takes me.
 
I think I have a proof involving finite descent; I eventually show that any ideal I is either equal to the ideal generated by some power of p (with 1 in the denominator), or we reach the point where the ideal generated by p is contained in I. Since I was able to show any proper ideal must be contained in (p), I am done... That is, any ideal is of the form (p^n) or the ring itself.

Sry for the lack of LaTex, I'm typing on a tablet.

Was there a more elegant approach?
 
Thanks very much, your hint was very concise and well chosen. All the best.
 
I am still confused. Could you please explain it in steps?
 
msg me if you want some hints
 
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or micromass for that matter, he was the architect