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I have read from Mike Prest's model theory for modules (London lecture note series) chapter 17 that a Ring of finite representation type has a decidable theory of modules. Here decidability was defined in the usual sense for theories : A theory is decidable if there is an effective method that decides whether the formula is in the theory or not.

I could not find a relevant literature or paper where this statement has been proved. I needed this result to assert a crucial proposition in my thesis. Any help would be greatly appreciated.

As a follow up question, are there specific papers investigating the syntactic completeness of the theory of modules over Artin Algebras? I can't seem to find much literature for this topic.

Thank you very much

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    $\begingroup$ The paper DECIDABILITY FOR THEORIES OF MODULES by FRANCHISE POINT AND MIKE PREST Journal of the London Mathematical SocietyVolume s2-38, Issue 2 p. 193-206 seems to indicate it is because the theories of finitely presented modules and all modules are the same. I guess the point must be that you can check truth in the indecomposables and there are only finitely many. $\endgroup$
    – Benjamin Steinberg
    Commented Nov 1, 2022 at 23:07
  • $\begingroup$ Thank you very much . The given reasoning makes complete sense to me. I will try to provide a proof in that direction. $\endgroup$
    – Yoneda Lemma
    Commented Nov 2, 2022 at 10:49

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