Abstract
Some years ago I published an example of an abelian group whose non-zero direct summands are all directly decomposable [2]; the group was obtained by the now standard technique of realising a suitable ring as an endomorphism ring. The present note extends the pathology and shows that such pathological groups occur even as direct summands of direct sums of indecomposable groups. Specifically, I shall indicate proofs of the following two theorems. (Throughout m is a fixed but arbitrary infinite cardinal.)
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
Manfred Dugas and Rüdiger Göbel, Torsion-free abelian groups with prescribed finitely topologized endomorphism rings, to appear in Proc. Amer. Math. Soc. 1983
A.L.S. Corner, Every countable reduced torsion-free ring is an endomorphism ring, Proc. London Math. Soc. 13 (1963), 687–710.
A.L.S. Corner, Endomorphism rings of torsion-free abelian groups, Proc. Internat. Conf. Theory Groups, 59–69, (New York, London, Paris, 1967).
L. Fuchs, Infinite Abelian Groups, Vol. II, ( New York, London, 1973 ).
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1983 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Corner, A.L.S. (1983). On the Existence of Very Decomposable Abelian Groups. In: Göbel, R., Lady, L., Mader, A. (eds) Abelian Group Theory. Lecture Notes in Mathematics, vol 1006. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-21560-9_18
Download citation
DOI: https://doi.org/10.1007/978-3-662-21560-9_18
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-12335-4
Online ISBN: 978-3-662-21560-9
eBook Packages: Springer Book Archive