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Abelian groups A such that the category of A-solvable groups is preabelian. (English) Zbl 0691.20038
Abelian group theory, Proc. 4th Conf., Perth/Aust. 1987, Contemp. Math. 87, 117-131 (1989).
[For the entire collection see Zbl 0666.00004.]
Let A be an abelian group with endomorphism ring E(A) operating on A from the left. For any abelian group G, the group Hom(A,G) carries a natural right E(A)-module structure which is induced by the composition of maps. Define $$\theta_ G: Hom(A,G)\otimes_{E(A)}A\to G$$ by $$\theta_ G(f\otimes a)=f(a)$$ for all $$f\in Hom(A,G)$$ and all $$a\in A$$. If $$\theta_ G$$ is an isomorphism G is said to be A-solvable. The author considers the category $${\mathcal T}_ A$$ of all A-solvable groups with $${\mathcal T}_ A$$-morphisms the group homomorphisms. Throughout, A is restricted to being a generalized rank 1 group (i.e. a reduced torsion- free abelian group with two-sided Noetherian and hereditary endomorphism ring) which is indecomposable. The categorical properties of $${\mathcal T}_ A$$ are investigated with particular emphasis on the question as to when $${\mathcal T}_ A$$ is preabelian. A complete solution is given: for groups of rank one this turns out to be the case precisely when A has idempotent type and, in this case, $${\mathcal T}_ A$$ is even abelian. If A has rank at least two and for no prime p the group of p-adic integers belongs to $${\mathcal T}_ A$$, then $${\mathcal T}_ A$$ is preabelian if and only if every $$G\in {\mathcal T}_ A$$ is A-torsion-free (i.e. $$G=Hom(A,G)A$$, and, for any $$f_ 1,...,f_ n\in Hom(A,G)$$, the group $$<f_ 1(A),...,f_ n(A)>$$ is A-projective (i.e. isomorphic to a direct summand of a direct sum of copies of A). The proofs make extensive use of the homological machinery introduced by D. M. Arnold and E. L. Lady in their fundamental paper on A-projective groups [Trans. Am. Math. Soc. 211, 225- 237 (1975; Zbl 0329.20033)]. The author includes several examples illustrating how the structure of A affects the category $${\mathcal T}_ A$$.
Reviewer: J.Hausen

##### MSC:
 20K20 Torsion-free groups, infinite rank 20K40 Homological and categorical methods for abelian groups 20K30 Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups 16S50 Endomorphism rings; matrix rings 20K15 Torsion-free groups, finite rank 18E05 Preadditive, additive categories