A TORSION-FREE ABELIAN GROUP EXISTS WHOSE QUOTIENT GROUP MODULO THE SQUARE SUBGROUP IS NOT A NIL-GROUP

The first example of a torsion-free abelian group $(A,+,0)$ such that the quotient group of $A$ modulo the square subgroup is not a nil-group is indicated (for both associative and general rings). In particular, the answer to the question posed by Stratton and Webb [‘Abelian groups, nil modulo a subgroup, need not have nil quotient group’,Publ. Math. Debrecen27(1980), 127–130] is given for torsion-free groups. A new method of constructing indecomposable nil-groups of any rank from $2$ to $2^{\aleph _{0}}$ is presented. Ring multiplications on $p$ -pure subgroups of the additive group of the ring of $p$ -adic integers are investigated using only elementary methods.