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A note on rank and direct decompositions of torsion-free Abelian groups. II

Published online by Cambridge University Press:  24 October 2008

A. L. S. Corner
A. L. S. Corner
Affiliation:
Worcester College, Oxford
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According to well-known theorems of Kaplansky and Baer–Kulikov–Kapla nsky–Fuchs (4, 2), the class of direct sums of countable Abelian groups and the class of direct sums of torsion-free Abelian groups of rank 1 are both closed under the formation of direct summands. In this note I give an example to show that the class of direct sums of torsion-free Abelian groups of finite rank does not share this closure property: more precisely, there exists a torsion-free Abelian group G which can be written both as a direct sum G = A⊕B of 2 indecomposable groups A, B of rank ℵ0 and as a direct sum G = ⊕n ε zCn of ℵ0 indecomposable groups Cn (nεZ) of rank 2, where Z is the set of all integers.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 66 , Issue 2 , September 1969 , pp. 239 - 240
Copyright
Copyright © Cambridge Philosophical Society 1969

References

REFERENCES

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(4)Kaplansky, I.Projective modules. Ann. of Math. 68 (1958), 372–377.CrossRefGoogle Scholar