Direct products and sums of torsion-free Abelian groups

Murley, C. E.

doi:10.1090/S0002-9939-1973-0311800-4

Direct products and sums of torsion-free Abelian groups
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by C. E. Murley

Proc. Amer. Math. Soc. 38 (1973), 235-241

DOI: https://doi.org/10.1090/S0002-9939-1973-0311800-4

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Abstract:

Let $A$ be a finite rank, indecomposable torsion-free Abelian group whose $p$-ranks are less than two for all primes $p$. Let $G$ be a direct product of copies of $A$, and $B$ be a nonzero countable pure subgroup of $G$ such that $B$ is the span of the homomorphic images of $A$ in $B$. Then it is shown that $B$ is a direct sum of copies of $A$. This result is applied to obtain a Krull-Schmidt theorem for direct sums of groups $A$ from a semirigid class of groups. In particular, if the groups $A$ have rank one, then the well-known BaerKulikov-Kaplansky theorem is obtained.

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