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DECIDABILITY FOR THEORIES OF MODULES OVER VALUATION DOMAINS

  • LORNA GREGORY (a1)

Abstract

Extending work of Puninski, Puninskaya and Toffalori in [5], we show that if V is an effectively given valuation domain then the theory of all V-modules is decidable if and only if there exists an algorithm which, given a, b ε V, answers whether a ε rad(bV). This was conjectured in [5] for valuation domains with dense value group, where it was proved for valuation domains with dense archimedean value group. The only ingredient missing from [5] to extend the result to valuation domains with dense value group or infinite residue field is an algorithm which decides inclusion for finite unions of Ziegler open sets. We go on to give an example of a valuation domain with infinite Krull dimension, which has decidable theory of modules with respect to one effective presentation and undecidable theory of modules with respect to another. We show that for this to occur infinite Krull dimension is necessary.

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Copyright

References

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[1]Fuchs, László and Salce, Luigi, Modules over non-Noetherian domains. Mathematical Surveys and Monographs, vol. 84, American Mathematical Society, Providence, RI, 2001.
[2]Gregory, Lorna, Ziegler Spectra of Valuation Rings, Ph.D. thesis, University of Manchester, 2011.
[3]Gregory, Lorna, Sobriety for the Ziegler spectrum of a Prüfer domain. Journal of Pure and Applied Algebra, vol. 217 (2013), no. 10, pp. 19801993.
[4]Herzog, Ivo, Elementary duality of modules. Transactions of the American Mathematical Society, vol. 340 (1993), no. 1, pp. 3769.
[5]Puninski, Gennadi, Puninskaya, Vera, and Toffalori, Carlo, Decidability of the theory of modules over commutative valuation domains. Annals of Pure and Applied Logic, vol. 145 (2007), no. 3, pp. 258275.
[6]Prest, Mike, Model theory and modules, London Mathematical Society Lecture Note Series, vol. 130, Cambridge University Press, Cambridge, 1988.
[7]Puninski, Gennadi, Cantor-Bendixson rank of the Ziegler spectrum over a commutative valuation domain, this Journal, vol. 64 (1999), no. 4, pp. 15121518.
[8]Puninski, Gennadi, Serial rings. Kluwer Academic Publishers, Dordrecht, 2001.
[9]Ziegler, Martin, Model theory of modules. Annals of Pure and Applied Logic, vol. 6 (1984), no. 2, pp. 149213.

Keywords

DECIDABILITY FOR THEORIES OF MODULES OVER VALUATION DOMAINS

  • LORNA GREGORY (a1)

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