Vol. 83, No. 2, 1979

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T as an 𝒢 submodule of G

William Jennings Wickless

Vol. 83 (1979), No. 2, 555–564

Let G be a mixed abelian group with torsion subgroup T. T is viewed as an submodule of G, where = End G. 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 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. T ess G, 2. I(G)I(T) as abelian groups, 3. Q⊈I(G). The condition T ess G 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 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 = End T.

Mathematical Subject Classification
Primary: 16A53, 16A53
Received: 24 July 1978
Revised: 6 November 1978
Published: 1 August 1979
William Jennings Wickless