On purity and quasi-equality of Abelian groups

We study the Cohn purity in an abelian group regarded as a left module over its endomorphism ring. We prove that if a finite rank torsion-free abelian group $G$ is quasiequal to a direct sum in which all summands are purely simple modules over their endomorphism rings then the module ${}_{E(G)}G$ is purely semisimple. This theorem makes it possible to construct abelian groups of any finite rank which are purely semisimple over their endomorphism rings and it reduces the problem of endopure semisimplicity of abelian groups to the same problem in the class of strongly indecomposable abelian groups.