ANNIHILATOR EQUIVALENCE OF TORSION-FREE ABELIAN-GROUPS

We define an equivalence relation on the class of torsion-free abelian groups under which two groups are equivalent if every pure subgroup of one has a non-zero image in the other, and each has a non-zero image in every torsion-free factor of the other. We study the closure properties of the equivalence classes, and the structural properties of the class of all equivalence classes. Finally we identify a class of groups which satisfy Krull-Schmidt and Jordan-Holder properties with respect to the equivalence.