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**Finite extensions of \(A\)-solvable Abelian groups.**
*(English)*
Zbl 0983.20055

Let \(A\) be an Abelian group and \(E=\text{End}_\mathbb{Z}(A)\) its endomorphism ring. Then there exist two natural functors between the category \({\mathcal A}b\) of Abelian groups and the category \({\mathcal M}_E\) of right \(E\)-modules which are related to \(A\) and \(E\), namely \(H_A=\operatorname{Hom}(A,-)\) and \(T_A=-\otimes_EA\). These functors from an adjoint pair and an Abelian group \(G\) is called (almost) \(A\)-solvable if the evaluation map \(\phi_G\colon T_AH_A(G)\to G\) is an (quasi-) isomorphism. The author studies the properties of almost \(A\)-solvable groups which obviously contain all torsion-free Abelian groups that are quasi-isomorphic to an \(A\)-solvable group. However, it is shown that the converse holds if \(A\) is flat and that the converse may fail if \(A\) is not flat. Let \(\mathcal{AC}_A\) be the category of all almost \(A\)-solvable groups and \(\mathcal{AM}_A\) be the category of all right \(E\)-modules such that the natural map \(\psi\colon M\to H_AT_A(M)\) is a quasi-isomorphism. Then \(H_A\) and \(T_A\) restrict to functors between \(\mathcal{AC}_A\) and \(\mathcal{AM}_A\) and two quasi-isomorphic Abelian groups \(A\) and \(B\) satisfy \(\mathcal{AC}_A=\mathcal{AC}_B\). Moreover, the class \(\mathcal{AM}_A\) is closed under submodules if and only if \(A\) is almost flat and has the quasi-Baer-splitting property. Finally, the author shows that two strongly indecomposable groups \(A\) and \(B\) of finite rank are quasi-isomorphic if and only if \(\mathcal{AC}_A=\mathcal{AC}_B\) which is a corollary of a more general result on determining the structure of \(A\) from \(\mathcal{AC}_A\).

Reviewer: Lutz Strüngmann (Jerusalem)

### MSC:

20K40 | Homological and categorical methods for abelian groups |

20K20 | Torsion-free groups, infinite rank |

### Keywords:

\(A\)-solvable groups; quasi-isomorphisms; categories of Abelian groups; categories of modules; endomorphism rings; torsionfree Abelian groups### References:

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