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

Published online by Cambridge University Press:  24 October 2008

A. L. S. Corner
A. L. S. Corner
Affiliation:
Corpus Christi College, Cambridge
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Extract

Fuchs((1), Problem 22) has asked the following question: Given positive integers ri (1 ≤ i ≤ 4) such thatr1 + r2 = r3 + r4, r1 ≠ r3, r1 ≠ r4, do there exist indecomposable torsion-free Abelian groups Gi (1 ≤ i ≤ 4) such that, where Gi is of rank ri (1 ≤ i ≤ 4)?* We shall show in this note that there do always exist groups Gi with the desired properties; in fact, we shall prove the following stronger result

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 57 , Issue 2 , April 1961 , pp. 230 - 233
Copyright
Copyright © Cambridge Philosophical Society 1961

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References

REFERENCES

(1)Fuchs, L., Abelian groups (Budapest, 1958).Google Scholar
(2)Fuchs, L., Notes on Abelian groups. I. Ann. Univ. Sci. Budapest, 2 (1959), 5–23.Google Scholar
(3)Jónsson, B., On direct decompositions of torsion-free Abelian groups. Math. Scand. 5 (1957), 230–5.CrossRefGoogle Scholar