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On the quasi-splitting of exact sequences. (English) Zbl 0749.20029

An (abelian) torsion-free group \(A\) has the Szele-property if every exact sequence \(0\to P\to F\) in which \(P\) and \(F\) are isomorphic to quasi- summands of finite direct sums of copies of \(A\), quasi-splits. The main result of the paper states that the Szele-property together with the condition \(\text{Tor}^ 1_{E(A)}(M,A)=0\) for all finitely related right \(E(A)\)-modules \(M\) with \(pd M\leq 1\) are equivalent to the fact that any f.g. proper left ideal of the quasi-endomorphism ring of \(A\) has a non-zero right annihilator. The structure of groups of finite rank with Szele-property is described and several examples concerning this property are presented.

MSC:

20K25 Direct sums, direct products, etc. for abelian groups
20K40 Homological and categorical methods for abelian groups
16S50 Endomorphism rings; matrix rings
20K15 Torsion-free groups, finite rank
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