Abstract
Let A be a torsion free abelian group and let G be the group of its automorphisms. Hallett and Hirsch proved that, if G is finite then G has to be a subdirect product of some copies of 6 explicitly distinguished small groups.
Our aim here is to extend this result to the case when G is periodic. We proceed in the context of rings and their groups of units
Supported by Polish research grant of KBN
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
D. M. Arnold, “Finite rank torsion free abelian groups and rings”, LNM 931, Springer-Verlag, 1982.
N. Bourbaki, “Algébre commutative” (Chs. 5 and 6), Hermann, Paris, 1964.
L. Fuchs, “Infinite abelian groups”, vol. 2, Academic Press, New York, 1973.
M. Hall, “The theory of groups”, Macmillan, New York, 1959.
J. T. Hallett, K. A. Hirsch, Torsion-free groups having finite automorphism groups, J. Algebra 2 (1965), 287–298.
J. T. Hallett, K. A. Hirsch, Die Konstruktion von Gruppen mit vorgeschriebenen Automorphismengruppen, J. Reine Angew. Math. 241 (1970), 32–46.
K. A. Hirsch, H. Zassenhaus, Finite automorphism groups of torsion-free groups, J. London Math. Soc. 41 (1966), 545–549.
A. A. Klein, A simple proof of a theorem on reduced rings, Canadian Math. Bull. 23 (1980), 495–496.
G. Lallement, “Semigroups and combinatorial applications”, Wiley-Interscience Publications, New York, 1979.
T. Y. Lam, “A first course in noncommutative rings”, Springer-Verlag, 1991.
W. Narkiewicz, “Elementary and analytic theory of algebraic numbers”, PWN and Springer-Verlag, Warszawa and Berlin, 1990.
D. S. Passman, “The algebraic structure of group rings”, Wiley-Interscience Publications, New York, 1977.
S. K. Sehgal, “Units in integral group rings”, Longman, Essex, 1993.
C. L. Siegel, Discontinuous groups, Annals of Math. 44 (1943), 674–689.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1995 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Krempa, J. (1995). Rings with Periodic Unit Groups. In: Facchini, A., Menini, C. (eds) Abelian Groups and Modules. Mathematics and Its Applications, vol 343. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-0443-2_26
Download citation
DOI: https://doi.org/10.1007/978-94-011-0443-2_26
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-4198-0
Online ISBN: 978-94-011-0443-2
eBook Packages: Springer Book Archive