To what extent is the structure theorem for finitely generated modules over principal ideal domains true over noncommutative domains? I'm in particular interested in noncommutative 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 finitelygenerated module over a noncommutative principal ideal domain is a direct sum of cyclic modules. 

