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