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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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  • $\begingroup$ Your starting point seems a bit naive, even for the infinite dihedral group and its group ring. $\endgroup$ Jan 14, 2013 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, 2013 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, 2013 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. $\endgroup$ Jan 15, 2013 at 9:33

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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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