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Endomorphism rings and a generalization of torsion-freeness and purity. (English) Zbl 0691.20040
A number of authors considered A-projective modules in the 70-ies. If A is a fixed abelian group, then A-projective modules are summands of direct sums of copies of A. The most interesting examples are \(A\subset {\mathbb{Q}}\) where we have good control over the endomorphism ring. Starting from this notion the present author considers the related notions A- torsion-freeness and A-purity which can be defined canonically via the A- socle of a group G, say \(S_ A(G)=<\phi (A):\) \(\phi \in Hom(A,G)>\). The author derives interesting relations between these notions and the endomorphism ring E(A). A byproduct are classical results, like some of Baer’s theorems on separable torsion-free abelian groups. The paper presents the major parts of the author’s Habilitationsschrift at Duisburg University (1986; Zbl 0634.20024).
Reviewer: R.Göbel

MSC:
20K30 Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups
16S50 Endomorphism rings; matrix rings
20K20 Torsion-free groups, infinite rank
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