An example of a noncommutative PID

Fred Byrd's suggestion of the quaternions does the trick, but if you want there to be some irreducible elements, you can use $\Bbb H[x]$, which is a left-and-right PID by the same argument you use for polynomial rings over fields. It is definitely not a division ring.

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As for a structure theorem, I believe the one you're looking for is for hereditary Noetherian prime rings, of which a left-and-right principal ideal domain is an example. Unfortunately I do not know this one by heart and don't have the resources handy.

If I were you I'd check out the work by Robson, Eisenbud, Griffith, Connell and Jategaonkar on the topic of hereditary Noetherian rings. I would guess the conclusion is probably available in Rowen's ring theory books and maybe Goodearl and Warfield's book on noncommutative noetherian rings.

I remember the theorem going something like this:

Hereditary Noetherian rings decompose into an Artinian ring and a finite product of hereditary Noetherian prime rings.

I also managed to find in Faith's Rings and things p 115 that proper homomorphic images of HNP rings are all Artinian serial rings. I highly suspect that these are exactly the Artinian rings that appear in the above decomposition in the Artinian part.

Let's compare this to Hungerford's result. (Principal ideal ring=PIR, principal ideal domain-PID, and special principal ideal ring=SPIR.)

Theorem 1: Every PIR is a finite direct sum of homomorphic images of PIDs.

Lemma 10: Every PIR is a (finite) direct sum of PIDs and SPIRs.

Corollary 11: Every SPIR is a homomorphic image of a PID.

Now a SPIR with a unique maximal ideal is an Artinian ring whose ideals are linearly ordered. A finite product of SPIRs is an example of an Artinian serial ring. So, that is why So, I sincerely believe that the hereditary Noetherian ring theorem is the proper extension of Hungerford's theorem.

The quaternions, I believe, will do the trick.

As for a general structure theorem, I'm afraid that's a bit out of my depth.