Puninski, G.; Puninskaya, V.; Toffalori, C. Decidability of the theory of modules over commutative valuation domains. (English) Zbl 1111.03011 Ann. Pure Appl. Logic 145, No. 3, 258-275 (2007). 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. Reviewer: S. R. Kogalovskij (Ivanovo) Cited in 1 ReviewCited in 9 Documents MSC: 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 Keywords:theory of modules; commutative valuation domain; decidability; Ziegler spectrum PDF BibTeX XML Cite \textit{G. Puninski} et al., Ann. Pure Appl. Logic 145, No. 3, 258--275 (2007; Zbl 1111.03011) Full Text: DOI References: [1] Eklof, P.C.; Fischer, E., Elementary theory of abelian groups, Ann. math. logic, 4, 115-171, (1972) · Zbl 0248.02049 [2] Facchini, A., Relative injectivity and pure-injective modules over Prüfer rings, J. algebra, 110, 380-406, (1987) · Zbl 0629.13008 [3] Fuchs, L.; Salce, L., () [4] Prest, M., () [5] M. Prest, Decidability for modules — summary, unpublished notes, 1991 [6] Puninski, G., Cantor – bendixson rank of the ziegler spectrum over a commutative valuation domain, J. symbolic logic, 64, 1512-1518, (1999) · Zbl 0961.03036 [7] Puninski, G., Serial rings, (2001), Kluwer · Zbl 1032.16001 [8] Szmielew, W., Elementary properties of abelian groups, Fund. math., 41, 203-271, (1955) · Zbl 0064.00803 [9] Ziegler, M., Model theory of modules, Ann. pure appl. logic, 26, 149-213, (1984) · Zbl 0593.16019 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.