Extensions of abelian groups of finite rank
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- by S. A. Khabbaz and E. H. Toubassi PDF
- Proc. Amer. Math. Soc. 50 (1975), 115-120 Request permission
Abstract:
Every abelian group $X$ of finite rank arises as the middle group of an extension $e:0 \to F \to X \to T \to 0$ where $F$ is free of finite rank $n$ and $T$ is torsion with the $p$-ranks of $T$ finite for all primes $p$. Given such a $T$ and $F$ we study the equivalence classes of such extensions which result from stipulating that two extensions ${e_i}:0 \to F \to {X_i} \to T \to 0,i = 1,2$, are equivalent if ${e_1} = \beta {e_2}\alpha$ for $\alpha \in \operatorname {Aut} (T)$ and $\beta \in \operatorname {Aut} (F)$. We reduce the problem to $T$ $p$-primary of finite rank, where in the one case $T$ is injective, and in the other case $T$ is reduced. Suppose $T = \Pi _{i = 1}^m{T_i}$. In our main theorems we prove that in each case these equivalence classes of extensions are in 1-1 correspondence with the equivalence classes of $n$-generated subgroups of $E$ where $E = \Pi _{i = 1}^m{E_i},{E_i} = \operatorname {End} ({T_i})$. Two $n$-generated subgroups of $E$ will be called equivalent if one can be mapped onto the other by an automorphism of $E$.References
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L. Fuchs, Infinite abelian groups. Vols. I, II, Pure and Appl. Math., vol. 36, Academic Press, New York, 1970, 1973. MR 41 #333.
- S. A. Khabbaz and E. H. Toubassi, The module structure of $\textrm {Ext}\ (F,\,T)$ over the endomorphism ring of $T$, Pacific J. Math. 54 (1974), 169β176. MR 360759
- S. A. Khabbaz and E. H. Toubassi, $\textrm {Ext}(A,\,T)$ as a module over $\textrm {End}(T)$, Proc. Amer. Math. Soc. 48 (1975), 269β275. MR 360865, DOI 10.1090/S0002-9939-1975-0360865-4
Additional Information
- © Copyright 1975 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 50 (1975), 115-120
- MSC: Primary 20K35
- DOI: https://doi.org/10.1090/S0002-9939-1975-0372073-1
- MathSciNet review: 0372073