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Some model theory of modules. III. On infiniteness of sets definable in modules

Published online by Cambridge University Press:  12 March 2014

Philipp Rothmaler
Philipp Rothmaler*
Affiliation:
Institut für Mathematik, Akademie der Wissenschaften der Ddr 1080 Berlin, East Germany
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Extract

This is the third and last part of an investigation in several topics of first order model theory of modules which I began in Some model theory of modules. I. On total transcendence of modules (this Journal, vol. 48 (1983), pp. 570–574) and continued in Some model theory of modules. II. On stability and categoricity of flat modules (this Journal, vol. 48 (1983), pp. 970–985). Throughout I refer to these papers as “Part I” and “Part II”.

Although these parts are only loosely connected, I will tacitly use the notation and preliminaries already introduced in the preceding ones. Further details are given in §0. Concerning Part II, the reader is assumed to be familiar with most of §1, a very small part of §4, and the criterion for elementary equivalence of modules over regular rings given in §3 (Lemma 20) of that part.

Using those tools in this paper I consider completeness of the (elementary) theory of all modules (and of some of its extensions) (§2), and eliminability of cardinality quantifiers in the elementary theories of modules (§3).

§2 is almost entirely devoted to a new short proof based on the technique developed in Part II of a theorem of Tukavkin which seems to be the first relevant result concerning the completeness of the theory of all modules (after the simple observation that any theory of infinite vector spaces is complete).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 49 , Issue 1 , March 1984 , pp. 32 - 46
Copyright
Copyright © Association for Symbolic Logic 1984

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References

REFERENCES

[BA1]Baudisch, A., Elimination of the quantifier Qα in the theory of abelian groups, Bulletin de l'Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 24 (1976), pp. 543–549.Google Scholar
[BA2]Baudisch, A., Theorien von Klassen abelscher Gruppen mit verallgemeinerten Quantoren, Dissertation B, Humboldt University, Berlin, 1977.Google Scholar
[BA3]Baudisch, A., Magidor-Malitz quantifiers in modules, this Journal, vol. 49 (1984), pp. 1–8.Google Scholar
[BL]Baldwin, J. T. and Lachlan, A. H., On strongly minimal sets, this Journal, vol. 36 (1971), pp. 79–96.Google Scholar
[BSTW]Baudisch, A., Seese, D., Tuschik, H.-P. and Weese, M., Decidability and generalized quantifiers, Akademie-Verlag, Berlin, 1980.Google Scholar
[ES]Eklof, P. and Sabbagh, G., Model-completions and modules, Annals of Mathematical logic, vol. 2 (1971), pp. 251–295.CrossRefGoogle Scholar
[FA]Faith, C., Algebra. Vols. I, II, Springer-Verlag, Berlin, 1973, 1976.Google Scholar
[LA]Lambek, J., Lectures on rings and modules, Blaisdell, Waltham, Massachusetts, 1966.Google Scholar
[MA]Macintyre, A., On ω1-categorical theories of abelian groups, Fundamenta Mathematicae, vol. 70 (1971), pp. 253–270.CrossRefGoogle Scholar
[MO]Morita, K., On S-rings in the sense of F. Kasch, Nagoya Mathematical Journal, vol. 27 (1966), pp. 688–695.CrossRefGoogle Scholar
[SA]Sabbagh, G., Sous-modules purs, existentiellement clos et élémentaires, Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences. Séries A et B, vol. 272 (1971), pp. A1289–A1292.Google Scholar
[SE]Sabbagh, G. and Eklof, P., Definability problems for modules and rings, this Journal, vol. 36 (1971), pp. 623–649.Google Scholar
[TÚ]Túkavkin, L. V., Modél'naá polnota téorii moduléj, preprint.Google Scholar
[ZN]Zalesskd, A. E. and Neroslavskij, O. M., Suščétvuút prostyé nélérovy kol'ca c détitel'ámi nulá, no béz idémpoténtov, Communications in Algebra, vol. 5 (1977), pp. 231–244.Google Scholar