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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 \({\mathcal T}_ A\).

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 |