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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
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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

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6. Reid, J. D., On the endomorphism rings of abelian p-groups (to appear).Google Scholar
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