Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-15T07:40:57.856Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Splitting of Modules and Abelian Groups

Published online by Cambridge University Press:  20 November 2018

Paul Hill
Paul Hill*
Affiliation:
Florida State University, Tallahassee, Florida
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In a fundamental paper on torsion-free abelian groups, R. Baer [1] proved that the group P of all sequences of integers with respect to componentwise addition is not free. This means precisely that P is not a direct sum of infinite cyclic groups. However, E. Specker proved in [9] that P has the property that any countable subgroup is free. Since an infinite abelian group G is called -free if each subgroup of rank less than is free, these results are equivalent to: P is -free but not free.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 26 , Issue 1 , February 1974 , pp. 68 - 77
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Baer, R., Abelian groups without elements of finite order, Duke Math. J. 3 (1937), 68–122.Google Scholar
2. Fuchs, L. and Rangaswamy, K. M., Finite-valued functions into a module (preprint).Google Scholar
3. Griffith, P., Infinite abelian group theory (University of Chicago Press, Chicago, 1970).Google Scholar
4. Griffith, P., A note on a theorem of Hill, Pacific J. Math. 29 (1969), 279–284.Google Scholar
5. Hill, P., The purification of subgroups of abelian groups, Duke Math. J. 88 (1970), 523–527.Google Scholar
6. Hill, P., On the freeness of abelian groups: a generalization of Pontryagins theorem, Bull. Amer. Math. Soc. 76 (1970), 1118–1120.Google Scholar
7. Kaup, L. and Keane, M. S., Induktive Limiten endlich erzeugter freier Moduln, Manuscripta Math. 1 (1969), 9–21.Google Scholar
8. Nöbeling, G., Verallgemeinerung eines Satzes von Herrn Specker, Invent. Math. 6 (1968), 41–55.Google Scholar
9. Specker, E., Additive Gruppen von Folgen ganzer Zahlen, Portugal. Math. 9 (1950), 131–140.Google Scholar
10. Weinberg, C., Free lattice-ordered abelian groups, Math. Ann. 151 (1963), 187–189.Google Scholar