On the Square Subgroup of a Mixed SI-Group

• Published in: Proceedings of the Edinburgh Mathematical Society Vol. 61; no. 1; pp. 295 - 304
• Main Authors: Andruszkiewicz, R. R., Woronowicz, M.
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.02.2018
• Subjects: Rings (mathematics); Subgroups; 17D99; SI-groups; Primary 20K99; the square subgroup of an abelian group; non-splitting abelian groups; Secondary 16D25; additive groups of rings
• ISSN: 0013-0915; 1464-3839
• DOI: 10.1017/S0013091517000165

The relation between the structure of a ring and the structure of its additive group is studied in the context of some recent results in additive groups of mixed rings. Namely, the notion of the square subgroup of an abelian group, which is a generalization of the concept of nil-group, is considered mainly for mixed non-splitting abelian groups which are the additive groups only of rings whose all subrings are ideals. A non-trivial construction of such a group of finite torsion-free rank no less than two, for which the quotient group modulo the square subgroup is not a nil-group, is given. In particular, a new class of abelian group for which an old problem posed by Stratton and Webb has a negative solution, is indicated. A new, far from obvious, application of rings in which the relation of being an ideal is transitive, is obtained.