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On the Square Subgroup of a Mixed SI-Group

Part of: Abelian groups Other nonassociative rings and algebras Modules, bimodules and ideals

Published online by Cambridge University Press:  15 February 2018

R. R. Andruszkiewicz  and
M. Woronowicz
R. R. Andruszkiewicz
Affiliation:
Institute of Mathematics, University of Białystok, 15-245 Białystok, K. Ciołkowskiego 1M, Poland (randrusz@math.uwb.edu.pl; mworonowicz@math.uwb.edu.pl)
M. Woronowicz*
Affiliation:
Institute of Mathematics, University of Białystok, 15-245 Białystok, K. Ciołkowskiego 1M, Poland (randrusz@math.uwb.edu.pl; mworonowicz@math.uwb.edu.pl)
*
*Corresponding author.
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Abstract

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.

Keywords

non-splitting abelian groupsadditive groups of ringsSI-groupsthe square subgroup of an abelian group

MSC classification

Primary: 20K99: None of the above, but in this section
Secondary: 16D25: Ideals 17D99: None of the above, but in this section
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 1 , February 2018 , pp. 295 - 304
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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