Almost completely decomposable torsion free abelian groups
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- by E. L. Lady PDF
- Proc. Amer. Math. Soc. 45 (1974), 41-47 Request permission
Abstract:
A finite rank torsion free abelian group $G$ is almost completely decomposable if there exists a completely decomposable subgroup $C$ with finite index in $G$. The minimum of $[G:C]$ over all completely decomposable subgroups $C$ of $G$ is denoted by $i(G)$. An almost completely decomposable group $G$ has, up to isomorphism, only finitely many summands. If $i(G)$ is a prime power, then the rank 1 summands in any decomposition of $G$ as a direct sum of indecomposable groups are uniquely determined. If $G$ and $H$ are almost completely decomposable groups, then the following statements are equivalent: (i) $G \oplus L \approx H \oplus L$ for some finite rank torsion free abelian group $L$. (ii) $i(G) = i(H)$ and $H$ contains a subgroup $G’$ isomorphic to $G$ such that $[H:G’]$ is finite and prime to $i(G)$. (iii) $G \oplus L \approx H \oplus L$ where $L$ is isomorphic to a completely decomposable subgroup with finite index in $G$.References
- D. M. Arnold and E. L. Lady, Endomorphism rings and direct sums of torsion free abelian groups, Trans. Amer. Math. Soc. 211 (1975), 225–237. MR 417314, DOI 10.1090/S0002-9947-1975-0417314-1
- László Fuchs, Infinite abelian groups. Vol. I, Pure and Applied Mathematics, Vol. 36, Academic Press, New York-London, 1970. MR 0255673
Additional Information
- © Copyright 1974 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 45 (1974), 41-47
- MSC: Primary 20K15
- DOI: https://doi.org/10.1090/S0002-9939-1974-0349873-6
- MathSciNet review: 0349873