Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-23T01:19:21.087Z Has data issue: false hasContentIssue false
  • English
  • Français

DIAGRAMS OF AN ABELIAN GROUP

Part of: Abelian groups

Published online by Cambridge University Press:  06 July 2009

THEODORE G. FATICONI
THEODORE G. FATICONI*
Affiliation:
Department of Mathematics, Fordham University, Bronx, NY 10458, USA (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we characterize quadratic number fields possessing unique factorization in terms of the power cancellation property of torsion-free rank-two abelian groups, in terms of Σ-unique decomposition, in terms of a pair of point set topological properties of Eilenberg–Mac Lane spaces, and in terms of the sequence of rational primes. We give a complete set of topological invariants of abelian groups, we characterize those abelian groups that have the power cancellation property in the category of abelian groups, and we characterize those abelian groups that have Σ-unique decomposition. Our methods can be used to characterize any direct sum decomposition property of an abelian group.

Keywords

abelian groupspoint set topological spaceshomologyhomotopyrings and modulescategoriesfunctors

MSC classification

Secondary: 20K99: None of the above, but in this section
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 80 , Issue 1 , August 2009 , pp. 38 - 64
Copyright
Copyright © Australian Mathematical Society 2009

References

[1] Arnold, D. M., Finite Rank Abelian Groups and Rings, Lecture Notes in Mathematics, 931 (Springer, New York, 1982).CrossRefGoogle Scholar
[2] Arnold, D. M. and Lady, L., ‘Endomorphism rings and direct sums of torsion-free abelian groups’, Trans. Amer. Math. Soc. 211 (1975), 225–237.CrossRefGoogle Scholar
[3] Eklof, P. C. and Mekler, A. H., Almost Free Modules. Set Theoretic Methods (North-Holland, Amsterdam, 1990).Google Scholar
[4] Faticoni, T. G., ‘A duality for self-slender modules’, Comm. Algebra 35(12) (2007), 4175–4182.CrossRefGoogle Scholar
[5] Faticoni, T. G., ‘Class number of an abelian group’, J. Algebra 314 (2007), 978–1008.CrossRefGoogle Scholar
[6] Faticoni, T. G., Direct Sum Decompositions of Torsion-free Finite Rank Groups (Chapman and Hall, New York/CRC, Boca Raton, FL, 2007).CrossRefGoogle Scholar
[7] Faticoni, T. G., Modules over Endomorphism Rings, Encyclopedia of Mathematics and its Applications, 130 (Cambridge University Press, Cambridge, 2009), to appear.CrossRefGoogle Scholar
[8] Faticoni, T. G., ‘Modules and point set topological spaces’, in: Abelian Groups, Rings, Modules, and Homological Algebra, Lecture Notes in Pure and Applied Mathematics, 249 (eds. H. P. Goeters and O. M. G. Jenda) (Chapman and Hall, New York/CRC, Boca Raton, FL, 2006), pp. 87–105.CrossRefGoogle Scholar
[9] Faticoni, T. G., ‘Modules over endomorphism rings as homotopy classes’, in: Abelian Groups and Modules (eds. A. Facchini and C. Menini) (Kluwer Academic, Dordrecht, 1995), pp. 163–183.CrossRefGoogle Scholar
[10] Fuchs, L., Infinite Abelian Groups I (Academic Press, New York, 1970).Google Scholar
[11] Fuchs, L., Infinite Abelian Groups II (Academic Press, New York, 1973).Google Scholar
[12] Hatcher, A., Algebraic Topology (Cambridge University Press, Cambridge, 2004).Google Scholar
[13] Huber, M. and Warfield, R. B. Jr., Homomorphisms Between Cartesian Powers of an Abelian Group, Lecture Notes in Mathematics, 874 (Springer, Berlin, 1981), pp. 202–227.Google Scholar
[14] Rotman, J., An Introduction to Homological Algebra, Pure and Applied Mathematics, 85 (Academic Press, New York, 1979).Google Scholar
[15] Stewart, I. N. and Tall, D. O., Algebraic Number Theory, 2nd edn (Chapman and Hall, London/CRC, Boca Raton, FL, 1987).Google Scholar