Geodesic
Abelian Groups Quasi-Injective Over their Endomorphism Rings
Canadian journal of mathematics, Tome 24 (1972) no. 4, pp. 617-621

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
Poole, George D.; Reid, James D. Abelian Groups Quasi-Injective Over their Endomorphism Rings. Canadian journal of mathematics, Tome 24 (1972) no. 4, pp. 617-621. doi: 10.4153/CJM-1972-056-6
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