Abstract
In this paper, we consider modules over principal ideal domains R. The objects are free R- modules F with two distinguished pure submodules F 0 and F1 with F 0 ∩ F1 = 0 and bounded quotient F/(F 0 ⊕ F 1) and morphisms are the usual R-homomorphisms which preserve the distinguished submodules. This category is denoted by cRep2.R and its objects, we say the cR2-modules are denoted by F = (F, F0, F 1). The rank of a cR2-module F is the rank of the free R-module F. We will show that cR2 -@#@ modules are direct sums of indecomposable cR2-modules of rank 1 or 2. The infinite series of indecomposable cR2-modules is well-known and given explicitly after our Main Theorem 1.4. The result was first shown for cR2modules of finite rank in Arnold and Dugas [4], then for countable rank, using heavy machinery due to Hill and Megibben [25] in Files and Göbel [20]. Our proof for arbitrary rank is based on [20] and illustrates the importance of Hill’s notion of an axiom-3 family of modules. The Main Theorem is applied to a classification of Butler groups with two critical types. 1 2
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Albrecht, U., Hill, P. (1987): Butler groups of infinite rank and axiom 3. Czech. Math. J. 37 112, 293–309.
Arnold, D. (1973): A class of pure subgroups of completely decomposable abelian groups. Proc. Amer. Math. Soc. 41, 37–44.
Arnold, D. (1982): Finite rank torsion-free abelian groups and rings. Vol. 931, Lecture Notes in Math., Springer Berlin.
Arnold, D., Dugas, M. (1993): Butler groups with finite typesets and free groups with distinguished subgroups. Comm. Algebra 21, 1947–1982.
Arnold, D., Dugas, M. (1995): Representations of finite posets and near-isomorphism of finite rank Butler groups. Rocky Mountain J. Math. 25, 591–609
Arnold, D., Vinsonhaler, C. (1993): Finite rank Butler groups: A survey of recent results. In: Abelian Groups, Lecture Notes in Pure and Applied Mathematics, Vol. 146, Marcel Dekker, pp. 17–39.
Benabdallah, K., Ouldbeddi, M.A.: Finite essential extensions of torsion-free abelian groups. To appear.
Bican, L., Salce, L. (1983): Butler groups of infinite rank. In: Abelian Group Theory, Lecture Notes in Math. 1006, Springer, pp. 171–189.
Brenner, S. (1967): Endomorphism algebras of vector spaces with distinguished sets of subspaces. J. Algebra 6, 100–114.
Brenner, S. (1974): Decomposition properties of small diagrams of modules. Sympos. Math. 13, 127–141.
Brenner, S. (1974): On four subspaces of a vector space. J. Algebra 29, 587–599.
Butler, M.C.R. (1965): A class of torsion-free abelian groups of finite rank. Proc. London Math. Soc. 15, 680–698.
Butler, M.C.R. (1968): Torsion-free modules and diagrams of vector spaces. Proc. London Math. Soc. 18, 635–652.
Butler, M.C.R. (1987): Some almost split sequences in torsion-free abelian group theory. In: Abelian Group Theory, Gordon and Breach London, pp. 291–302.
Cohen, J., Gluck, H. (1970): Stacked bases for modules over principal ideal domains. J. Algebra 14, 493–505.
Corner, A.L.S., Göbel, R. (1985): Prescribing endomorphism algebras, a unified treatment. Proc. London Math. Soc. 50, 447–479.
Dugas, M., Göbel, R.: Applications of abelian groups and model theory to algebraic structures, pp. 41–62 in Infinite Groups 94, Walter de Gruyter, Berlin, New York 1995.
Dugas, M., Göbel, R.: Endomorphism rings of B2-groups of infinite rank, to appear in Israel Journ. 1997.
Dugas, M., Göbel, R., May, W.: Free modules with two distinguished submodules. To appear.
Files, S., Göbel, R. (1997): Gauβ′ theorem for two submodules, to appear in Math. Zeitschr.
Fuchs, L. (1970,1973): Infinite abelian groups, Vol. I, II, Academic Press New York.
Fuchs, L., Magidor, M. (1993): Butler groups of arbitrary cardinality. Israel Journ. Math. 84, 239–263.
Göbel, R., May, W. (1990): Four submodules suffice for realizing algebras over commutative rings. Journ. Pure and Appl. Algebra 65, 29–43.
Hill, P. (1985): Simultaneous decompositions of an abelian group and a subgroup. Quart. J. Math. Oxford (2) 36, 425–433.
Hill, P., Megibben, C. (1989): Generalizations of the stacked bases theorem. Trans. Amer. Math. Soc. 312, 377–402.
Hill, P. Megibben, C. (1986): Axiom 3 modules, Trans. Amer. Math. Soc. 295, 715–734.
Lady, E.L. (1974): Almost completely decomposable torsion-free abelian groups. Proc. Amer. Math. Soc. 45, 41–47.
Lady, E.L. (1975): Nearly isomorphic torsion-free abelian groups. J. Algebra 35, 235–238.
Lewis, W. (1993): Almost completely decomposable groups with two critical types. Comm. in Algebra 21(2), 607–614.
Mader, A. (1996): Almost completely decomposable abelian groups. Book in print at Gordon and Breach London.
Mader, A., Mutzbauer, O., Rangaswamy, K.M. (1994): A generalization of Butler groups. In: Abelian group theory and related topics, Contemporary Math. 171, Amer. Math. Soc. Providence, pp. 257–275.
Mader, A., Vinsonhaler, C. (1994): Classifying almost completely decomposable abelian groups. J. Algebra 170, 754–780.
Shelah, S. (1985): Incompactness in regular cardinals, Notre Dame Journal of Formal Logic 26, 195–228.
Simson, D. (1992): Linear Representations of Partially Ordered Sets and Vector Space Categories. Algebra, Logic and Applications, Vol. 4, Gordon and Breach Science Publishers London.
Vinsonhaler, C. (1996): Balanced Butler groups and representations, pp. 113–122 in: Abelian Groups, Proceed. Colorado Springs 1995, Kluwer Academic Publisher, New York 1996
Author information
Authors and Affiliations
Additional information
The first author is supported by Baylor University’s Summer Sabbatical Program and the second author by the project No. G-0294-081.06/93 of the German-Israeli Foundation for Scientific Research & Development
Rights and permissions
About this article
Cite this article
Dugas, M., Göbel, R. Classification of Modules with Two Distinguished Pure Submodules and Bounded Quotients. Results. Math. 30, 264–275 (1996). https://doi.org/10.1007/BF03322195
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF03322195