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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

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