Purifiability in pure subgroups

Abstract

Let $G$ be an abelian group. A subgroup $A$ of $G$ is said to be {\it purifiable} in $G$ if, among the pure subgroups of $G$ containing $A$, there exists a minimal one. Suppose that $A$ is purifiable in $G$ and $H$ is a pure subgroup of $G$ containing $A$. Then $A$ need not be purifiable in $H$. In this note, we ask for conditions that guarantee that $A$ is purifiable in the intermediate group $H$. First, we prove that if $A$ is a torsion--free purifiable subgroup of a group $G$ and $H$ is a direct summand of $G$ containing $A$, then $A$ is purifiable in $H$. Next, we characterize the pure subgroups $K$ of a group $G$ with the property that a torsion--free finite rank subgroup $A$ of $K$ is purifiable in $K$ if $A$ is purifiable in $G$.