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 sub...

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Bibliographic Details
Published inBulletin of the Australian Mathematical Society Vol. 94; no. 3; pp. 449 - 456
Main Authors ANDRUSZKIEWICZ, R. R., WORONOWICZ, M.
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 01.12.2016
Subjects
Group theory
Subgroups
abelian groups
nil-groups
primary 20K99
square subgroups
secondary 13A99
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Summary: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. Debrecen 27 (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.
ISSN:0004-9727
1755-1633
DOI:10.1017/S0004972716000435