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A TORSION-FREE ABELIAN GROUP EXISTS WHOSE QUOTIENT GROUP MODULO THE SQUARE SUBGROUP IS NOT A NIL-GROUP
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 94 / Issue 3 / December 2016
- Published online by Cambridge University Press:
- 30 August 2016, pp. 449-456
- Print publication:
- December 2016
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- Article
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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.
