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.)
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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 ).
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© 1983 Springer-Verlag Berlin Heidelberg
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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
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DOI: https://doi.org/10.1007/978-3-662-21560-9_18
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