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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
Affiliation:
Worcester College, Oxford

Extract

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.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1969

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References

REFERENCES

(1)Corner, A. L. S.A note on rank and direct decompositions of torsion-free Abelian groups. Proc. Cambridge Philos. Soc. 57 (1961), 230233.CrossRefGoogle Scholar
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(3)Jónsson, B.On direct decompositions of torsion-free Abelian groups. Math. Scand. 5 (1957), 230235.CrossRefGoogle Scholar
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