To what extent is the structure theorem for finitely generated modules over principal ideal domains true over non-commutative domains? I'm in particular interested in non-commutative euclidean domains especially the twisted polynomial ring $K\langle X \rangle$ over a field $K$ (i.e. such that $Xa = \sigma(a)X$ for some automorphism $\sigma$ of K).
The question is thoroughly explored in Chapter 3 of Nathan Jacobson's Theory of Rings. I took a quick look, and it looks like the analogous results go through in the noncommutative case. For example, Theorem 19 in Chapter 3 states that a finitely-generated module over a noncommutative principal ideal domain is a direct sum of cyclic modules.