It’s well known that when a ring R is a PID, every submodule of a free R-module is free. I’m interested in cases when the converse holds — that is, in rings R which have the property that every submodule of a free R-module is free. It’s easy enough to see that such a ring cannot have any zero divisors and that every ideal of the ring must be principal. I would like to know if there are any “noncommutative PIDs”, that is, if there are any rings which (1) have no zero divisors, (2) have only principal ideals, and (3) are noncommutative. Is there any sort of structure theorem for such rings? For instance, Hungerford proved in this paper that every principal ideal ring (a ring whose ideals are all principal) which is commutative is a direct sum of homomorphic images of PIDs.
EDIT: Robert Lewis reminded me that one class of examples of the ring I am looking for is division rings. Are there any examples that are not division rings? (After all, the ideal structure of a division ring is not very interesting.)
Fred Byrd’s suggestion of the quaternions does the trick, but if you want there to be some irreducible elements, you can use 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.
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.