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The construction of $$A$$-solvable Abelian groups. (English) Zbl 0823.20056
Let $$A$$ be a torsion-free abelian group and let $$E(A)$$ denote the endomorphism ring of $$A$$. Then $$A$$ is a left $$E(A)$$-module giving rise to the pair $$(\text{Hom}(A,-)\otimes_{E(A)}A)$$ of adjoint functors between the category of abelian groups and the category of right $$E(A)$$-modules. These functors have been used extensively in the investigation of torsion-free abelian groups originating with D. M. Arnold and E. L. Lady [Trans. Am. Math. Soc. 211, 225-237 (1975; Zbl 0329.20033)] and D. M. Arnold and C. E. Murley [Pac. J. Math. 56, 7-20 (1975; Zbl 0337.13010)], and continued in a series of papers by Albrecht and others.
The abelian group $$G$$ is defined to be $$A$$-solvable if the natural map $$\theta_ G : \text{Hom} (A,G) \otimes_{E(A)} A \to G$$ is an isomorphism. The supply of examples of $$A$$-solvable groups was limited: most examples were subgroups of direct products of copies of $$A$$, or severe restrictions were based on the structure of $$E(A)$$. Assuming various augmentations of the standard set theory ZFC (e.g. V=L, $$\nabla_{\kappa}$$), the author constructs a plethora of $$A$$-solvable groups $$G$$ such that $$\text{Hom} (G,A) = 0$$.
Reviewer: J.Hausen (Houston)

##### MSC:
 20K20 Torsion-free groups, infinite rank 20K40 Homological and categorical methods for abelian groups 16D90 Module categories in associative algebras 16S50 Endomorphism rings; matrix rings 20K25 Direct sums, direct products, etc. for abelian groups
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