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).

$\begingroup$ Your starting point seems a bit naive, even for the infinite dihedral group and its group ring. $\endgroup$ – Charles Matthews Jan 14 '13 at 14:10

$\begingroup$ I guess you interpret the group ring of $D_\infty$ as a twisted polynomial ring? Is the theory of f.g. modules over this group ring complicated? $\endgroup$ – YCor Jan 14 '13 at 15:22

$\begingroup$ @Charles, it would probably be more helpful to the OP if you explained your comment a little more. $\endgroup$ – MTS Jan 14 '13 at 16:57

$\begingroup$ OK, the representation theory of the infinite dihedral group over a field is a special case, in that the group ring is really twisted Laurent polynomials. You have to assume X acts invertibly, in other words. $\endgroup$ – Charles Matthews Jan 15 '13 at 9:33
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.