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# Pure-projective modules and positive constructibility

## Extract

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].

## References

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[Rot1]Rothmaler, Ph., Mittag-Leffler modules and positive atomicity, Habilitationsschrift, Kiel, 1994.
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[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. 445469.
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[W]Wisbauer, R., Foundations of module and ring theory, Gordon and Breach, Philadelphia, 1991.

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