# Structure of f.g. modules over a non-commutative ring

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).

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Your starting point seems a bit naive, even for the infinite dihedral group and its group ring. –  Charles Matthews Jan 14 '13 at 14:10
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? –  YCor Jan 14 '13 at 15:22
@Charles, it would probably be more helpful to the OP if you explained your comment a little more. –  MTS Jan 14 '13 at 16:57
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. –  Charles Matthews Jan 15 '13 at 9:33