Let G be a mixed abelian
group with torsion subgroup T. T is viewed as an ℰ submodule of G, where
ℰ =EndG. It is shown that T is superfluous in G if and only if, ∀p, either Tp
is divisible or G∕Tp is not p divisible. If G is not reduced, T is essential
in G if and only if T contains a Z(p∞). Let I(G)[I(T)] be the ℰ injective
hull of G[T]. Then I(G) = I(T) ⊕ X with X torsion free divisible and T is
a pure subgroup of I(G). This can be used to obtain several results; for
example, if Q⊈I(T), TFAE: 1. TessG, 2. I(G)≅I(T) as abelian groups, 3.
Q⊈I(G). The condition TessG is characterized if T is a summand or if
G is algebraically compact. If T is bounded or if T is a p-group, T1= (0)
and G is reduced cotorsion, T is not essential. In fact, for bounded T there
is an ℰ isomorphism I(G)≅I(T) ⊕ I(G∕T). Some information is obtained
on the p-basic subgroups of I(T) as a function of those of T. A condition
is given for I(T) ⊇⊕cQ. These last theorems specialize to IE(T), where
E =EndT.
