Extensions of abelian groups of finite rank
S. A. Khabbaz and E. H. Toubassi
Proc. Amer. Math. Soc. 50 (1975), 115-120
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Abstract: Every abelian group of finite rank arises as the middle group of an extension where is free of finite rank and is torsion with the -ranks of finite for all primes . Given such a and we study the equivalence classes of such extensions which result from stipulating that two extensions , are equivalent if for and . We reduce the problem to -primary of finite rank, where in the one case is injective, and in the other case is reduced. Suppose . 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 -generated subgroups of where . Two -generated subgroups of will be called equivalent if one can be mapped onto the other by an automorphism of .
L. Fuchs, Infinite abelian groups. Vols. I, II, Pure and Appl. Math., vol. 36, Academic Press, New York, 1970, 1973. MR 41 #333.
A. Khabbaz and E.
H. Toubassi, The module structure of
𝐸𝑥𝑡(𝐹,𝑇) over the endomorphism ring
of 𝑇, Pacific J. Math. 54 (1974),
169–176. MR 0360759
A. Khabbaz and E.
H. Toubassi, 𝐸𝑥𝑡(𝐴,𝑇)
as a module over 𝐸𝑛𝑑(𝑇), Proc. Amer. Math. Soc. 48 (1975), 269–275. MR 0360865
(50 #13312), http://dx.doi.org/10.1090/S0002-9939-1975-0360865-4
- 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 over the endomorphism ring of , Pacific J. Math. 54 (1975), 169-176. MR 0360759 (50:13206)
- -, as a module over , Proc. Amer. Math. Soc. 48 (1975), 269-275. MR 0360865 (50:13312)
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