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Pure-projective modules and positive constructibility

Published online by Cambridge University Press:  12 March 2014

T. G. Kucera  and
Ph. Rothmaler
T. G. Kucera
Affiliation:
Department of Mathematics, University of Manitoba, Winnipeg, Manitoba, CanadaR3T 2N2, E-mail: [email protected]
Ph. Rothmaler
Affiliation:
Institut Für Logik, Christian-Albrechts-Universitätzu Kiel, D-24098 Kiel, Germany, E-mail: [email protected] Institut Für Logik, Christian-Albrechts-Universitätzu Kiel, D-24098 Kiel, Germany, E-mail: [email protected]
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In modules many ‘positive’ versions of model-theoretic concepts turn out to be equivalent to concepts known in classical module theory—by ‘positive’ we mean that instead of allowing arbitrary first-order formulas in the model-theoretic definitions only positive primitive formulas are taken into consideration. (This feature is due to Baur's quantifier elimination for modules, cf. [Pr], however we will not make explicit use of it here.) Often this allows one to combine model-theoretic methods with algebraic ones. One instance of this is the result proved in [Rot1] (see also [Rot2]) that the Mittag-Leffler modules are exactly the positively atomic modules. This paper is parallel to the one just mentioned in that it is proved here, Theorem 3.1, that the pure-projective modules are exactly the positively constructible modules. The following parallel facts from module theory and from model theory led us to this result: every pure-projective module is Mittag-Leffler and the converse is true for countable (in fact even countably generated) modules, cf. [RG]; every constructible model is atomic and the converse is true for countable models, cf. [Pi].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 65 , Issue 1 , March 2000 , pp. 103 - 110
Copyright
Copyright © Association for Symbolic Logic 2000

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References

REFERENCES

[AF]Azumaya, G. and Facchini, A., Rings of pure global dimension zero and Mittag-Leffler modules, Journal of Pure and Applied Algebra, vol. 62 (1989), pp. 109–122.CrossRefGoogle Scholar
[JL]Jensen, C. U. and Lenzing, H., Model theoretic algebra, Gordon and Breach, New York, 1989.Google Scholar
[Kap]Kaplansky, I., Projective modules, Annals of Mathematics, vol. 68 (1958), pp. 372–377.CrossRefGoogle Scholar
[Pi]Pillay, A., An introduction to stability theory, Oxford Logic Guides, vol. 8, Oxford, 1983.Google Scholar
[Pr]Prest, M., Model theory and modules, London Mathematical Society Lecture Notes Series, vol. 130, Cambridge, 1988.CrossRefGoogle Scholar
[PPR]Puninski, G. E., Prest, M., and Rothmaler, Ph., Rings described by various purities, Communications in Algebra, vol. 27 (1999), pp. 2127–2162.CrossRefGoogle Scholar
[RG]Raynaud, M. and Gruson, L., CritÈres de platitude et de projectivitÉ, Seconde partie, Inventiones Mathematical vol. 13 (1971), pp. 52–89.CrossRefGoogle Scholar
[Rot1]Rothmaler, Ph., Mittag-Leffler modules and positive atomicity, Habilitationsschrift, Kiel, 1994.Google Scholar
[Rot2]Rothmaler, Ph., Mittag-Leffler modules, Proceedings of the Florence Conference on Model Theory and Algebra, Aug 20–25,1995, Annals of Pure and Applied Logic, vol. 88, 1997, pp. 227–239.Google Scholar
[Rot3]Rothmaler, Ph., Purity in model theory, Proceedings of the Conferences on Model Theory and Algebra, Essen/Dresden, 1994/95, Droste, M. und Göbel, R. (Hrsg.), Algebra, Logic and Application Series, Bd. 9, Gordon and Breach, 1997, pp. 445–469.Google Scholar
[War]Warfeld, R. B., Purity and algebraic compactness for modules, Pacific Journal of Mathematics, vol. 28 (1969), pp. 699–719.CrossRefGoogle Scholar
[W]Wisbauer, R., Foundations of module and ring theory, Gordon and Breach, Philadelphia, 1991.Google Scholar