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Direct product decomposition of theories of modules

Published online by Cambridge University Press:  12 March 2014

Steven Garavaglia
Steven Garavaglia*
Affiliation:
University of California, Berkeley, California 94720
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Extract

This paper is mainly concerned with describing complete theories of modules by decomposing them (up to elementary equivalence) into direct products of simpler modules. In §1, I give a decomposition theorem which works for arbitrary direct product theories T. Given such a T, I define T-indecomposable structures and show that every model of T is elementarily equivalent to a direct product of T-indecomposable models of T. In §2, I show that if R is a commutative ring then every R-module is elementarily equivalent to ΠMM where M ranges over the maximal ideals of R and M is the localization of at M. This is applied to prove that if R is a commutative von Neumann regular ring and TR is the theory of R-modules then the TR-indecomposables are precisely the cyclic modules of the form R/M where M is a maximal ideal. In §3, I use the decomposition established in §2 to characterize the ω1-categorical and ω-stable modules over a countable commutative von Neumann regular ring and the superstable modules over a commutative von Neumann regular ring of arbitrary cardinality. In the process, I also prove several general characterizations of ω-stable and superstable modules; e.g., if R is any countable ring, then an R-moduIe is ω-stable if and only if every R-module elementarily equivalent to it is equationally compact.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 44 , Issue 1 , March 1979 , pp. 77 - 88
Copyright
Copyright © Association for Symbolic Logic 1979

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References

REFERENCES

[1]Baur, W., Elimination of quantifiers for modules (to appear).Google Scholar
[2]Cohen, I.S. and Kaplansky, I., Rings for which every module is a direct sum of cyclic modules, Mathematische Zeitschrift, vol. 54 (1951), pp. 97–101.CrossRefGoogle Scholar
[3]Eklof, P., Some model theory of abelian groups, this Journal, vol. 37 (1972), pp. 335–341.Google Scholar
[4]Eklof, P. and Fisher, E., The elementary theory of abelian groups, Annals of Mathematical Logic, vol. 4 (1972), pp. 115–171.CrossRefGoogle Scholar
[5]Eklof, P. and Sabbagh, G., Model-completions and modules, Annals of Mathematical Logic, vol. 2 (1971), pp. 251–295.CrossRefGoogle Scholar
[6]Feferman, S. and Vaught, R., The first order properties of products of algebraic systems, Fundamenta Mathematicae, vol. 97 (1959), pp. 57–103.CrossRefGoogle Scholar
[7]Galvin, F., Horn sentences, Annals of Mathematical Logic, vol. 1(1970), pp. 389–422.CrossRefGoogle Scholar
[8]Garavaglia, S., Elementary equivalence of modules, unpublished.Google Scholar
[9]Gilmer, R., Multiplicative ideal theory, Marcel Dekker, New York, 1972.Google Scholar
[10]Kaplansky, I., Commutative rings, Allyn and Bacon, Boston, Massachusetts, 1970.Google Scholar
[11]Lang, S., Algebra, Addison-Wesley, Reading, Massachusetts, 1971.Google Scholar
[12]McDonald, B., Finite rings with identity, Marcel Dekker, New York, 1974.Google Scholar
[13]Macintyre, A., On ω1-categorical theories of abelian groups, Fundamenta Mathematicae, vol. 120 (1971), pp. 253–270.CrossRefGoogle Scholar
[14]Martyanov, V.I., The theory of abelian groups with predicates specifying a subgroup and with endomorphism operations, Algebra and Logic, vol. 14(1976), pp.330–334.CrossRefGoogle Scholar
[15]Monk, L., Elementary-recursive decision procedures, Ph.D. dissertation, University of California, Berkeley, 1975.Google Scholar
[16]Northcott, D.G., An introduction to homological algebra, Cambridge University Press, London and New York, 1966.Google Scholar
[17]Sabbagh, G., Aspects logique de la pureté dans les modules, Comptes Rendus Hebdomadaires des Séances de l'Academie des Sciences, Serie A, vol. 271 (1970), pp. 909–912.Google Scholar