It is a well-known fact that modules over a commutative ring in general cannot be classified, and it is also well-known that we have to impose severe restrictions on either the ring or on the class of modules to solve this problem. One of the restrictions on the modules comes from freeness assumptions which have been intensively studied in recent decades. Two interesting, distinct but typical examples are the papers by Blass [1] and Eklof [8], both jointly with Shelah. In the first case the authors...

Let $\lambda$ be an infinite cardinal. Set ${\lambda }_0=\lambda$, define ${\lambda }_{i+1}=2^{{\lambda }_i}$ for every $i=0,1,\cdots$, take $\mu$ as the first cardinal with ${\lambda }_i<\mu$, $i=0,1,\cdots$ and put $\kappa ={\left({\mu }^{{\aleph }_0}\right)}^{+}$. If $F$ is a torsion-free group of cardinality at least $\kappa$ and $K$ is its subgroup such that $F/K$ is torsion and $|F/K|\le \lambda$, then $K$ contains a non-zero subgroup pure in $F$. This generalizes the result from a previous paper dealing with $F/K$$p$-primary.