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DECIDABILITY OF MODULES OVER A BÉZOUT DOMAIN D+XQ[X] WITH D A PRINCIPAL IDEAL DOMAIN AND Q ITS FIELD OF FRACTIONS

Published online by Cambridge University Press:  17 April 2014

GENA PUNINSKI  and
CARLO TOFFALORI
GENA PUNINSKI
Affiliation:
DEPARTMENT OF MATHEMATICS, BELARUSIAN STATE UNIVERSITY, PRASPEKT NEZALEZHNOSTI 4, MINSK 220030, BELARUSE-mail:[email protected]
CARLO TOFFALORI
Affiliation:
UNIVERSITY OF CAMERINO, SCHOOL OF SCIENCE AND TECHNOLOGIES, DIVISION OF MATHEMATICS, VIA MADONNA DELLE CARCERI 9, 62032 CAMERINO, ITALYE-mail:[email protected]
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Abstract

We describe the Ziegler spectrum of a Bézout domain B=D+XQ[X] where D is a principal ideal domain and Q is its field of fractions; in particular we compute the Cantor–Bendixson rank of this space. Using this, we prove the decidability of the theory of B-modules when D is “sufficiently” recursive.

Keywords

03C6013F0513F30Bézout domaindecidabilityZiegler spectrumCantor–Bendixson rankKrull–Gabriel dimension
Type
Articles
Information
The Journal of Symbolic Logic , Volume 79 , Issue 1 , March 2014 , pp. 296 - 305
Copyright
Copyright © Association for Symbolic Logic 2014 

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References

REFERENCES

Fuchs, L. and Salce, L., Modules over non-noetherian domains, Mathematical Surveys and Monographs Series 84, American Mathematical Society, Providence, RI, 2001.Google Scholar
Gregory, L., Decidability for theories of modules over valuation domains. Preprint, available at www.math.uni-konstanz.de/∼gregory/Decmodulesgregory.pdf.Google Scholar
Prest, M., Model theory and modules, London Mathematical Society Lecture Note Series 130, Cambridge University Press, Cambridge, 1988.Google Scholar
Prest, M., Decidability for modules—summary, unpublished notes, 1991.Google Scholar
Prest, M., Purity, spectra and localization, Encyclopedia of Mathematics and its Applications 121, Cambridge University Press, Cambridge, 2009.Google Scholar
Puninski, G., Serial rings, Kluwer, Dordrecht, 2001.Google Scholar
Puninski, G., Prest, M., and Rothmaler, P., Rings described by various purities. Communications in Algebra, vol. 27 (1999), pp. 2127–2162.Google Scholar
Puninski, G., Puninskaya, V., and Toffalori, C., Decidability of the theory of modules over commutative valuation domains. Annals of Pure and Applied Logic, vol. 145 (2007), pp. 258–275.Google Scholar
Puninski, G. and Toffalori, C., Some model theory of modules over Bézout domains. The width. Preprint, available at http://www.logique.jussieu.fr/modnet/Publications/Preprint%20server/papers/639/639.pdf.CrossRefGoogle Scholar
Ziegler, M., Model theory of modules. Annals of Pure and Applied Logic, vol. 26 (1984),pp. 149–213.Google Scholar