Abstract
The study of pairs of modules (over a Dedekind domain) arises from two different perspectives, as a starting step in the analysis of tuples of submodules of a given module, or also as a particular case in the analysis of Abelian structures made by two modules and a morphism between them. We discuss how these two perspectives converge to pairs of modules, and we follow the latter one to obtain an alternative approach to the classification of pairs of torsionfree objects. Then we restrict our attention to pairs of free modules. Our main results are that the theory of pairs of free Abelian groups is co-recursively enumerable, and that a few remarkable extensions of this theory are decidable.
Similar content being viewed by others
References
Baur, W.: Undecidability of the theory of Abelian groups with a subgroup. Proc. Amer. Math. Soc. 55, 125–129 (1976)
Baur, W.: On the elementary theory of quadruples of vectorspaces. Ann. Math. Logic 19, 243–262 (1980)
L. Bican - R. El Bashir - E.: Enochs, All modules have flat covers. Bull. London Math. Soc. 33, 385–390 (2001)
Enochs, E.: Torsion free covering modules. Proc. Amer. Math. Soc. 14, 884–889 (1963)
Herzog, I.: Elementary duality of modules. Trans. Amer. Math. Soc. 340, 37–69 (1993)
Hill, P., Megibben, C.: Generalizations of the stacked bases theorem. Trans. Amer. Math. Soc. 312, 377–402 (1989)
Kozlov, G.T., Kokorin, A.I.: The elementary theory of Abelian groups without torsion with a predicate selecting a subgroup. Algebra i Logika 8, 320–334 (1969); English translation in Algebra and Logic 8, 182–190 (1969)
Prest, M.: Model theory and representation type of algebras. Logic Colloquium '86, Studies in Logic, North Holland, Amsterdam, 1988, pp. 266–280
Prest, M.: Model theory and modules, London Math. Soc. Lecture Notes Ser. 130, Cambridge University Press, Cambridge, 1988
Point, F., Prest, M.: Decidability for theories of modules. J. London Math. Soc. 38, 193–206 (1988)
Schmitt, P.H.: The elementary theory of torsionfree Abelian groups with a predicate specifying a subgroup. Z. Math. Logik Grundl. Math. 28, 323–329 (1982)
Xu, J.: Flat covers of modules, Lecture Notes in Mathematics 1634, Springer, Berlin, 1996
Ziegler, M.: Model theory of modules. Ann. Pure Applied Logic 26, 149–213 (1984)
Author information
Authors and Affiliations
Corresponding author
Additional information
Work performed under the auspices of GNSAGA-INDAM
Rights and permissions
About this article
Cite this article
Cittadini, S., Toffalori, C. On pairs of free modules over a Dedekind domain. Arch. Math. Logic 45, 75–95 (2006). https://doi.org/10.1007/s00153-005-0311-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00153-005-0311-1