Faithful abelian groups of infinite rank

Albrecht, Ulrich

doi:10.1090/S0002-9939-1988-0938637-8

Faithful abelian groups of infinite rank
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by Ulrich Albrecht

Proc. Amer. Math. Soc. 103 (1988), 21-26

DOI: https://doi.org/10.1090/S0002-9939-1988-0938637-8

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Abstract:

Let $B$ be a subgroup of an abelian group $G$ such that $G/B$ is isomorphic to a direct sum of copies of an abelian group $A$. For $B$ to be a direct summand of $G$, it is necessary that $G$ be generated by $B$ and all homomorphic images of $A$ in $G$. However, if the functor $\operatorname {Hom} (A, - )$ preserves direct sums of copies of $A$, then this condition is sufficient too if and only if $M{ \otimes _{E(A)}}A$ is nonzero for all nonzero right $E(A)$-modules $M$. Several examples and related results are given.

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