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DECIDABILITY OF THE THEORY OF MODULES OVER PRÜFER DOMAINS WITH INFINITE RESIDUE FIELDS

Published online by Cambridge University Press:  21 December 2018

LORNA GREGORY ,
SONIA L’INNOCENTE ,
GENA PUNINSKI  and
CARLO TOFFALORI
LORNA GREGORY
Affiliation:
SCHOOL OF SCIENCE AND TECHNOLOGIES DIVISION OF MATHEMATICS UNIVERSITY OF CAMERINO VIA MADONNA DELLE CARCERI 9 CAMERINO 62032, ITALYE-mail: [email protected]
SONIA L’INNOCENTE
Affiliation:
SCHOOL OF SCIENCE AND TECHNOLOGIES DIVISION OF MATHEMATICS UNIVERSITY OF CAMERINO VIA MADONNA DELLE CARCERI 9 CAMERINO 62032, ITALYE-mail: [email protected]
GENA PUNINSKI
Affiliation:
FACULTY OF MECHANICS AND MATHEMATICS BELARUSIAN STATE UNIVERSITY AV. NEZALEZHNOSTI 4 MINSK 220030, BELARUS
CARLO TOFFALORI
Affiliation:
SCHOOL OF SCIENCE AND TECHNOLOGIES DIVISION OF MATHEMATICS UNIVERSITY OF CAMERINO VIA MADONNA DELLE CARCERI 9 CAMERINO 62032, ITALYE-mail: [email protected]
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Abstract

We provide algebraic conditions ensuring the decidability of the theory of modules over effectively given Prüfer (in particular Bézout) domains with infinite residue fields in terms of a suitable generalization of the prime radical relation. For Bézout domains these conditions are also necessary.

Keywords

Primary 03C6003C9803B2513F05Prüfer domainBézout domainprime radical relationdecidability
Type
Articles
Information
The Journal of Symbolic Logic , Volume 83 , Issue 4 , December 2018 , pp. 1391 - 1412
Copyright
Copyright © The Association for Symbolic Logic 2018 

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References

REFERENCES

Brewer, J. W., Conrad, P. F., and Montgomery, P. R., Lattice-ordered groups and a conjecture for adequate domains. Proceedings of the American Mathematical Society, vol. 43 (1974), pp. 31–35.CrossRefGoogle Scholar
Eklof, P. and Herzog, I., Model theory of modules over a serial ring. Annals of Pure and Applied Logic, vol. 72 (1995), pp. 145–176.CrossRefGoogle Scholar
Fuchs, L. and Salce, L., Modules Over Non-Noetherian Domains, Mathematical Surveys and Monographs, vol. 84, American Mathematical Society, Providence, RI, 2001.Google Scholar
Gilmer, R., Multiplicative Ideal Theory, Queen’s Papers in Pure and Applied Mathematics, vol. 90, Queen’s University, Kingston, Ontario, 1992.Google Scholar
Gregory, L., Decidability for theories of modules over valuation domains, this Journal, vol. 80 (2015), pp. 684–711.Google Scholar
Heitmann, R. C., Generating ideals in Prüfer domains. Pacific Journal of Mathematics, vol. 62 (1976), pp. 117–126.CrossRefGoogle Scholar
Herzog, I., Elementary duality of modules. Transactions of the American Mathematical Society, vol. 340 (1993), pp. 37–69.CrossRefGoogle Scholar
L’Innocente, S. and Point, F., Bézout domains and lattice ordered modules, preprint, 2018, arXiv:1604.05922 [math.LO].Google Scholar
L’Innocente, S., Point, F., Puninski, G., and Toffalori, C., The Ziegler spectrum of the ring of entire complex valued functions, this Journal, to appear.Google Scholar
L’Innocente, S., Toffalori, C., and Puninski, G., On the decidability of the theory of modules over the ring of algebraic integers. Annals of Pure and Applied Logic, vol. 168 (2017), pp. 1507–1516.CrossRefGoogle Scholar
Prest, M., Model Theory and Modules, London Mathematical Society Lecture Notes Series, vol. 130, Cambridge University Press, Cambridge, 1990.Google Scholar
Prest, M., Purity, Spectra and Localization, Encyclopedia of Mathematics and its Applications, vol. 121, Cambridge University Press, Cambridge, 2009.CrossRefGoogle Scholar
Puninski, G., The Krull-Gabriel dimension of a serial ring. Communications in Algebra, vol. 31 (2003), pp. 5977–5993.CrossRefGoogle 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.CrossRefGoogle Scholar
Puninski, G. and Toffalori, C., Decidability of modules over a Bézout domain $D + XQ[X]$ with D a principal ideal domain and Q its field of fractions, this Journal, vol. 79 (2014), pp. 296–305.Google Scholar
Puninski, G. and Toffalori, C., Some model theory of modules over Bézout domains. The Width, Journal of Pure and Applied Algebra, vol. 219 (2015), pp. 807–829.CrossRefGoogle Scholar
Tuganbaev, A., Distributive rings, uniserial rings of fractions and endo-Bezout modules. Journal of Mathematical Sciences, vol. 114 (2003), pp. 1185–1203.CrossRefGoogle Scholar
Ziegler, M., Model theory of modules. Annals of Pure and Applied Logic, vol. 26 (1984), pp. 149–213.CrossRefGoogle Scholar