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Towards the decidability of the theory of modules over finite commutative rings. (English) Zbl 1168.03024
The reviewer conjectured that if \(R\) is a sufficiently recursively given ring then the theory of \(R\)-modules is decidable iff \(R\) is of tame representation type. This paper begins with an excellent summary of the history of work on the problem and the current state of knowledge. It should also be recalled that the decidability problem for modules over finite rings effectively contains that for the variety generated by a finite universal algebra.
The authors concentrate on the case that \(R\) is a finite commutative ring, without loss of generality local, but a good deal of their careful style of analysis applies much more generally. They make essential use of the Klingler-Levy analysis of finitely generated modules over commutative noetherian rings and they note that, in particular, this gives a clear distinction between the tame and wild cases in this context. They clearly present the interpretations involved and give a careful analysis of the information required on the Ziegler spectrum to prove decidability.
They are able to establish complete results, supporting the conjecture, except in the case that \(R\) is a pullback of two finite valuation rings over a field (the Gelfand-Ponomarev algebras are long-standing examples of this open case).
This is a clever and exemplary paper which might well fulfil the authors’ wish to inspire further work on the problem.

MSC:
03C60 Model-theoretic algebra
03B25 Decidability of theories and sets of sentences
13L05 Applications of logic to commutative algebra
16D50 Injective modules, self-injective associative rings
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
16P10 Finite rings and finite-dimensional associative algebras
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