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Many books and papers on the representation theory of finite dimensional algebras state that the first order theory for finite dimensional modules for the free algebra on two generators is undecidable because one can interpret the word problem for finitely presented groups in this theory (I am working over a field). Then they cite some source. The references I have checked (some I cannot access) all reduce the word problem to the theory of all modules. The proof they give clearly does not work for finite dimensional modules. It can be saved by using the undecidability of the uniform word problem for FINITE groups (or semigroups) and Malcev's theorem on residual finiteness of linear (semi)groups.

Question: can somebody give me a source where the theorem is really proved for finite dimensional modules?

Update: The paper http://attach.sciencedirect.com/science/article/pii/S0168007297000250 mentions the undecidability for f.g. modules using the uniform word problem for finite groups but it gives the impression it is folklore and gives no source.

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  • $\begingroup$ The sources I looked at were Baur and the book of Prest on Model Theory and Modules. $\endgroup$
    – Benjamin Steinberg
    Commented Aug 6, 2012 at 21:52
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    $\begingroup$ If Mark Sapir or Olga Kharlampovich do not come here and answer the question personally, I would suggest hunting down their notes on algorithmic problems in algebra. The answer might not be there, but it is the first place I would look. Gerhard "Ask Me About System Design" Paseman, 2012.08.06 $\endgroup$
    – Gerhard Paseman
    Commented Aug 6, 2012 at 23:16
  • $\begingroup$ Gerhard, I didn't see it in the survey. $\endgroup$
    – Benjamin Steinberg
    Commented Aug 7, 2012 at 11:51

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