## 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.## References

- Reinhold Baer,
*Abelian groups without elements of finite order*, Duke Math. J.**3**(1937), no. 1, 68–122. MR**1545974**, DOI 10.1215/S0012-7094-37-00308-9 - Bernard Charles,
*Sous-groupes fonctoriels et topologies*, Studies on Abelian Groups (Symposium, Montpellier, 1967) Springer, Berlin, 1968, pp. 75–92 (French). MR**0240195** - D. W. Dubois,
*Cohesive groups and $p$-adic integers*, Publ. Math. Debrecen**12**(1965), 51–58. MR**220823**, DOI 10.5486/pmd.1965.12.1-4.07 - L. Fuchs,
*Abelian groups*, Publishing House of the Hungarian Academy of Sciences, Budapest, 1958. MR**0106942** - László Fuchs,
*Infinite abelian groups. Vol. I*, Pure and Applied Mathematics, Vol. 36, Academic Press, New York-London, 1970. MR**0255673** - Bjarni Jónsson,
*On direct decomposition of torsion free Abelian groups*, Math. Scand.**7**(1959), 361–371. MR**122872**, DOI 10.7146/math.scand.a-10582 - Irving Kaplansky,
*Projective modules*, Ann. of Math. (2)**68**(1958), 372–377. MR**0100017**, DOI 10.2307/1970252 - George Kolettis,
*Homogeneously decomposable modules*, Studies on Abelian Groups (Symposium, Montpellier, 1967) Springer, Berlin, 1968, pp. 223–238. MR**0244228** - L. Ya. Kulikov,
*On direct decompositions of groups*, Amer. Math. Soc. Transl. (2)**2**(1956), 23–87. MR**0075951**, DOI 10.1090/trans2/002/02 - C. E. Murley,
*The classification of certain classes of torsion free Abelian groups*, Pacific J. Math.**40**(1972), 647–655. MR**322077** - Fred Richman,
*A class of rank-$2$ torsion free groups*, Studies on Abelian Groups (Symposium, Montpellier, 1967) Springer, Berlin, 1968, pp. 327–333. MR**0244380** - R. B. Warfield Jr.,
*A Krull-Schmidt theorem for infinite sums of modules*, Proc. Amer. Math. Soc.**22**(1969), 460–465. MR**242886**, DOI 10.1090/S0002-9939-1969-0242886-2 - R. B. Warfield Jr.,
*An isomorphic refinement theorem for Abelian groups*, Pacific J. Math.**34**(1970), 237–255. MR**266994**

## Bibliographic Information

- © Copyright 1973 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**38**(1973), 235-241 - MSC: Primary 20K15
- DOI: https://doi.org/10.1090/S0002-9939-1973-0311800-4
- MathSciNet review: 0311800