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

[ttzhou]

Homework Statement

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.

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 $R_D$ 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.

 

[micromass]

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.

 

[ttzhou]

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

 

[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?

 

[micromass]

That's indeed what I had in mind.

 

[ttzhou]

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

 

[arya1234]

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

 

[ttzhou]

msg me if you want some hints

 

[ttzhou]

or micromass for that matter, he was the architect

 

[micromass]

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

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