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.