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

  1. Jun 6, 2012 #1
    1. The problem statement, all variables and given/known data

    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.

    2. Relevant equations

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

    3. 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.
     
    Last edited: Jun 6, 2012
  2. jcsd
  3. Jun 6, 2012 #2

    micromass

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    Ok, can you show that an element x that is not invertible in [itex]R_D[/itex] can be written as py.

    In general, can you show that if x is not invertible, then it can be written as [itex]p^ny[/itex] with y invertible.
     
  4. Jun 6, 2012 #3
    Thank you for the hint! Will see where it takes me.
     
  5. Jun 7, 2012 #4
    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?
     
  6. Jun 7, 2012 #5

    micromass

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    That's indeed what I had in mind.
     
  7. Jun 7, 2012 #6
    Thanks very much, your hint was very concise and well chosen. All the best.
     
  8. Jun 7, 2012 #7
    I am still confused. Could you please explain it in steps?
     
  9. Jun 7, 2012 #8
    msg me if you want some hints
     
    Last edited by a moderator: Jun 14, 2012
  10. Jun 7, 2012 #9
    or micromass for that matter, he was the architect
     
  11. Jun 7, 2012 #10

    micromass

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    You will have to show us what you tried. Did you prove my post #2?
     
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