Direct sums and refinement. (English) Zbl 0920.20045
A set \(A\) of objects in an additive category has the refinement property if every direct summand of a direct sum of objects in \(A\) is isomorphic to a direct sum of objects of \(A\). This situation is frequently considered in the theory of torsion-free abelian groups of finite rank, in which the refinement property can be used as a substitute for the Krull-Schmidt Theorem or Kaplansky’s Test Problems.
The author proves that various classes of finite rank torsion-free abelian groups have the refinement property in categories whose isomorphisms are near- or quasi-isomorphisms of abelian groups.
The author proves that various classes of finite rank torsion-free abelian groups have the refinement property in categories whose isomorphisms are near- or quasi-isomorphisms of abelian groups.
Reviewer: Ph.Schultz (Perth)
MSC:
| 20K25 | Direct sums, direct products, etc. for abelian groups |
| 20K15 | Torsion-free groups, finite rank |
| 20K40 | Homological and categorical methods for abelian groups |
Keywords:
additive categories; near-isomorphisms; quasi-isomorphisms; direct sums; finite rank torsion-free Abelian groups; refinement propertyReferences:
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