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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
up vote 3 down vote accepted

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.

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Thanks. That's what I'm looking for. – user26756 Jan 16 '13 at 21:22

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