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On the Square Subgroup of a Mixed SI-Group

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

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 ([email protected]; [email protected])
M. Woronowicz*
Affiliation:
Institute of Mathematics, University of Białystok, 15-245 Białystok, K. Ciołkowskiego 1M, Poland ([email protected]; [email protected])
*
*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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References

1 Aghdam, A. M., Square subgroup of an Abelian group, Acta. Sci. Math. 51 (1987), 343–348.Google Scholar
2 Aghdam, A. M. and Najafizadeh, A., Square subgroups of rank two Abelian groups, Colloq. Math. 117(1) (2009), 19–28.CrossRefGoogle Scholar
3 Aghdam, A. M. and Najafizadeh, A., Square submodule of a module, Mediterr. J. Math. 7(2) (2010), 195–207.CrossRefGoogle Scholar
4 Andruszkiewicz, R. R., The classification of integral domains in which the relation of being an ideal is transitive, Comm. Algebra 31 (2003), 2067–2093.CrossRefGoogle Scholar
5 Andruszkiewicz, R. R. and Sobolewska, M., Accessible subrings and Kurosh's chains of associative rings, J. Aust. Math. Soc. 95(2) (2013), 145–157.CrossRefGoogle Scholar
6 Andruszkiewicz, R. R. and Woronowicz, M., On SI-groups, Bull. Aust. Math. Soc. 91(1) (2015), 92–103.CrossRefGoogle Scholar
7 Andruszkiewicz, R. R. and Woronowicz, M., Some new results for the square subgroup of an Abelian group, Comm. Algebra 44(6) (2016), 2351–2361.CrossRefGoogle Scholar
8 Balcerzyk, S. and Józefiak, T., Commutative Noetherian and Krull rings (PWN – Polish Scientific Publishers, Warsaw, 1989).Google Scholar
9 Călugăreanu, G., Breaz, S., Modoi, C., Pelea, C. and Vălcan, D., Exercises in Abelian group theory (Kluwer Academic Publishers, Dordrecht, 2003).CrossRefGoogle Scholar
10 Feigelstock, S., Additive groups of rings whose subrings are ideals, Bull. Austral. Math. Soc. 55 (1997), 477–481.CrossRefGoogle Scholar
11 Feigelstock, S., Additive groups of rings, Volume 1 (Pitman Advanced Publishing Program, Boston, 1983).Google Scholar
12 Feigelstock, S., Additive groups of rings, Volume 2 (Research Notes in Mathematics, Volume 169, Longman, London, 1988).Google Scholar
13 Fuchs, L., Ringe und ihre additive Gruppe, Publ. Math. Debrecen 4 (1956), 488–508.CrossRefGoogle Scholar
14 Fuchs, L., Infinite Abelian groups, Volume 1 (Academic Press, New York, 1970).Google Scholar
15 Fuchs, L., Infinite Abelian groups, Volume 2 (Academic Press, New York, 1973).Google Scholar
16 Fuchs, L. and Rangaswamy, K. M., On generalized regular rings, Math. Z. 107 (1968), 71–81.CrossRefGoogle Scholar
17 Kruse, R. L., Rings in which all subrings are ideals, Canad. J. Math. 20 (1968), 862–871.CrossRefGoogle Scholar
18 Najafizadeh, A., On the square submodule of a mixed module, Gen. Math. Notes 27(1) (2015), 1–8.Google Scholar
19 Rédei, L., Vollidealringe im weiteren Sinn. I, Acta Math. Acad. Sci. Hungar. 3 (1952), 243–268.CrossRefGoogle Scholar
20 Stratton, A. E. and Webb, M. C., Abelian groups, nil modulo a subgroup, need not have nil quotient group, Publ. Math. Debrecen. 27 (1980), 127–130.CrossRefGoogle Scholar