# Finitely generated modules over non-commutative principal ideal rings

The structure theorem for finitely generated modules over a principal ideal domain is well known. My question is about the noncommutative version of this theorem:

Let $R$ be a ring with identity without non-zero zero-divisors and such that every left ideal and every right ideal of $R$ is principal. An example of such a ring is a division ring $D$ or a polynomial ring $D[x]$ over the division ring $D$. If $M$ is a finitely generated module over $R$ is there a simple structure for $M$ as in the commutative case?

• What ideals are principal: left, right, two-sided, all...? – user26857 Sep 30 '15 at 10:43
• the left and the right ideals are both principal. – user166934 Sep 30 '15 at 13:05
• I don't recall where (if) the proof breaks down. I think a proof in the noncommutative case might appear in P. M. Cohn's book but I can't see the whole thing to verify. He is talking specifically about noncommutative PIDs. – rschwieb Sep 30 '15 at 17:42