Hostname: page-component-76fb5796d-2lccl Total loading time: 0 Render date: 2024-04-30T03:23:23.737Z Has data issue: false hasContentIssue false
  • English
  • Français

Abelian Groups Quasi-Injective Over their Endomorphism Rings

Published online by Cambridge University Press:  20 November 2018

George D. Poole  and
James D. Reid
George D. Poole
Affiliation:
Texas Tech University, Lubbock, Texas
James D. Reid
Affiliation:
Wesley an University, Middletown, Connecticut
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.

L. Fuchs has posed the problem of identifying those abelian groups that can serve as the additive structure of an injective module over some ring [1, p. 179], and in particular of identifying those abelian groups which are injective as modules over their endomorphism rings [1, p. 112]. Richman and Walker have recently answered the latter question, generalized in a non-trivial way [7], and have shown that the groups in question are of a rather restricted structure.

In this paper we consider abelian groups which are quasi-injective over their endomorphism rings. We show that divisible groups are quasi-injective as are direct sums of cyclic p-groups. Quasi-injectivity of certain direct sums (products) is characterized in terms of the summands (factors). In general it seems that the answer to the question of whether or not a group G is quasinjective over its endomorphism ring E depends on how big HomE(H, G) is, with H a fully invariant subgroup of G.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 4 , August 1972 , pp. 617 - 621
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Fuchs, L., Infinite abelian groups, Vol. 1 (Academic Press, New York, 1970).Google Scholar
2. Hill, P., Endomorphism rings generated by units, Trans. Amer. Math. Soc. 141 (1969), 99–105.Google Scholar
3. Kaplansky, I., Infinite abelian groups (University of Michigan Press, Ann Arbor, 1969).Google Scholar
4. Lambek, J., Lectures on rings and modules (Blaisdell Publishing Company, Waltham, 1966).Google Scholar
5. Nunke, R. J., Homology and direct sums of countable abelian groups, Math. Z. 101 (1967), 182–212.Google Scholar
6. Reid, J. D., On the endomorphism rings of abelian p-groups (to appear).Google Scholar
7. Richman, F. and Walker, E. A., Modules over p.i.d.'s that are infective over their endomorphism rings (to appear in Proceedings of the Park City Ring Theory Conference).Google Scholar