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)


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
Full Text: EuDML


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