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

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
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