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.


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
Full Text: DOI


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