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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Direct products and sums of torsion-free Abelian groups


Author: C. E. Murley
Journal: Proc. Amer. Math. Soc. 38 (1973), 235-241
MSC: Primary 20K15
MathSciNet review: 0311800
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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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DOI: http://dx.doi.org/10.1090/S0002-9939-1973-0311800-4
PII: S 0002-9939(1973)0311800-4
Keywords: $ p$-rank of a group, $ J$-group, strongly homogeneous group, semirigid class of groups, cohesive group
Article copyright: © Copyright 1973 American Mathematical Society