Direct decompositions of torsion-free Abelian groups of finite rank

Abstract

A certain wide new class of torsion-free Abelian groups of finite rank, which admits an exhaustive description of all its decompositions into direct sums of indecomposable summands, is constructed. It is proved that if n=r1+...+rs=l 1+...+l t are two decompositions of the number n into a sum of natural numbers such that all the terms of the first decomposition do not exceed n−t+1 while all the terms of the second decomposition do not exceed n−s+1, then there exists an Abelian group of rank n which admits a decomposition into a direct sum of indecomposable groups of ranks zl,..., zs and a decomposition into a direct sum of indecomposable groups of ranks ll,..., lt.