On pureness in Abelian groups

Torsion-free Abelian groups $G$ and $H$ are called quasi-equal ($G\approx H$) if $\lambda G\subset H\subset G$ for a certain natural number $\lambda$. It is known (see [3]) that the quasi-equality of torsion-free Abelian groups can be represented as the equality in an appropriate factor category. Thus while dealing with certain group properties it is usual to prove that the property under consideration is preserved under the transition to a quasi-equal group. This trick is especially frequently used when the author investigates module properties of Abelian groups, here a group is considered as a left module over its endomorphism ring. On the other hand, an actual problem in the Abelian group theory is a problem of investigation of pureness in the category of Abelian groups (see [1]). We consider the pureness introduced by P. Cohn [5] for Abelian groups as modules over their endomorphism rings. The feature of the investigation of the properties of pureness for the Abelian group $G$ as the module $_{E(G)}G$ lies in the fact that this is a more general situation than the investigation of pureness for a unitary module over an arbitrary ring $R$ with the identity element. Indeed, if $_R M$ is an arbitrary unitary left module and $M^+$ is its Abelian group, then each element from $R$ can be identified with an appropriate endomorphism from the ring $E(M^+)$ under the canonical ring homomorphism $R\to E(M^+)$. Then it holds that if $_{E(M^+)}N$ is a pure submodule in $_{E(M^+)}M^+$, then $_R N$ is a pure submodule in $_R M$. In the present paper the interrelations between pureness, servantness, and quasi-decompositions for Abelian torsion-free groups of finite rank will be investigated.