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

1. Jun 6, 2012

### ttzhou

1. The problem statement, all variables and given/known data

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

Show that if $p$ is a small prime and $D = R \setminus \left\langle p\right\rangle$ then $R_D$ 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 $R_D$ 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. Jun 6, 2012

### micromass

Staff Emeritus
Ok, can you show that an element x that is not invertible in $R_D$ can be written as py.

In general, can you show that if x is not invertible, then it can be written as $p^ny$ with y invertible.

3. Jun 6, 2012

### ttzhou

Thank you for the hint! Will see where it takes me.

4. Jun 7, 2012

### ttzhou

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?

5. Jun 7, 2012

### micromass

Staff Emeritus
That's indeed what I had in mind.

6. Jun 7, 2012

### ttzhou

Thanks very much, your hint was very concise and well chosen. All the best.

7. Jun 7, 2012

### arya1234

I am still confused. Could you please explain it in steps?

8. Jun 7, 2012

### ttzhou

msg me if you want some hints

Last edited by a moderator: Jun 14, 2012
9. Jun 7, 2012

### ttzhou

or micromass for that matter, he was the architect

10. Jun 7, 2012

### micromass

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