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Tensor Products and the Splitting of Abelian Groups

Published online by Cambridge University Press:  20 November 2018

D. A. Lawver  and
E. H. Toubassi
D. A. Lawver
Affiliation:
University of Arizona, Tucson, Arizona
E. H. Toubassi
Affiliation:
University of Arizona, Tucson, Arizona
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In [2], Irwin, Khabbaz, and Rayna discuss the splitting problem for abelian groups through the use of the tensor product. Throughout the paper they make a basic assumption, namely, that the torsion subgroup contains but one primary component. Under this restriction they introduce the concept of “splitting length”, which is a positive integer indicator of how far a group is from splitting. The results obtained along these lines may be extended to groups whose torsion subgroups contain any finite number of primary components by applying the work of Oppelt [4].

Irwin, Khabbaz, and Rayna [2] define the notion of a p-sequence and show that for groups A where T(A) is p-primary and A/T(A) has rank one, the existence of a torsion-free element with a p-sequence is sufficient for the group to split.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 5 , October 1971 , pp. 764 - 770
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Fuchs, L., Notes on abelian groups. I, Ann. Univ. Sci. Budapest 2 (1959), 5–23.Google Scholar
2. Irwin, J. M., Khabbaz, S. A., and Rayna, G., The role of the tensor product in the splitting oj abelian groups, J. Algebra U (1970), 423–442.Google Scholar
3. Kaplansky, I., Infinite abelian groups, revised edition (University of Michigan Press, Ann Arbor, 1968).Google Scholar
4. Oppelt, J. A., Mixed abelian groups, Can. J. Math. 19 (1967), 1259–1262.Google Scholar
5. Toubassi, E. H., On the splitting of mixed groups of torsion-free rank one (to appear).Google Scholar