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.)