Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-24T04:40:05.517Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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