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Decidability of the theory of modules over commutative valuation domains. (English) Zbl 1111.03011
A commutative ring \(V\) with unity is said to be a valuation ring if the lattice of ideals of \(V\) is a chain. A valuation ring without zero divisors is called a valuation domain. It is proved that if \(V\) is an effectively given valuation domain such that its value group is dense and archimedean, then the theory of all (unitary) \(V\)-modules is decidable.

03B25 Decidability of theories and sets of sentences
13A18 Valuations and their generalizations for commutative rings
13G05 Integral domains
13C99 Theory of modules and ideals in commutative rings
03C60 Model-theoretic algebra
Full Text: DOI
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