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On the Square Subgroup of a Mixed SI-Group

  • R. R. Andruszkiewicz (a1) and M. Woronowicz (a1)
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.

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1 Aghdam, A. M., Square subgroup of an Abelian group, Acta. Sci. Math. 51 (1987), 343348.
2 Aghdam, A. M. and Najafizadeh, A., Square subgroups of rank two Abelian groups, Colloq. Math. 117(1) (2009), 1928.
3 Aghdam, A. M. and Najafizadeh, A., Square submodule of a module, Mediterr. J. Math. 7(2) (2010), 195207.
4 Andruszkiewicz, R. R., The classification of integral domains in which the relation of being an ideal is transitive, Comm. Algebra 31 (2003), 20672093.
5 Andruszkiewicz, R. R. and Sobolewska, M., Accessible subrings and Kurosh's chains of associative rings, J. Aust. Math. Soc. 95(2) (2013), 145157.
6 Andruszkiewicz, R. R. and Woronowicz, M., On SI-groups, Bull. Aust. Math. Soc. 91(1) (2015), 92103.
7 Andruszkiewicz, R. R. and Woronowicz, M., Some new results for the square subgroup of an Abelian group, Comm. Algebra 44(6) (2016), 23512361.
8 Balcerzyk, S. and Józefiak, T., Commutative Noetherian and Krull rings (PWN – Polish Scientific Publishers, Warsaw, 1989).
9 Călugăreanu, G., Breaz, S., Modoi, C., Pelea, C. and Vălcan, D., Exercises in Abelian group theory (Kluwer Academic Publishers, Dordrecht, 2003).
10 Feigelstock, S., Additive groups of rings whose subrings are ideals, Bull. Austral. Math. Soc. 55 (1997), 477481.
11 Feigelstock, S., Additive groups of rings, Volume 1 (Pitman Advanced Publishing Program, Boston, 1983).
12 Feigelstock, S., Additive groups of rings, Volume 2 (Research Notes in Mathematics, Volume 169, Longman, London, 1988).
13 Fuchs, L., Ringe und ihre additive Gruppe, Publ. Math. Debrecen 4 (1956), 488508.
14 Fuchs, L., Infinite Abelian groups, Volume 1 (Academic Press, New York, 1970).
15 Fuchs, L., Infinite Abelian groups, Volume 2 (Academic Press, New York, 1973).
16 Fuchs, L. and Rangaswamy, K. M., On generalized regular rings, Math. Z. 107 (1968), 7181.
17 Kruse, R. L., Rings in which all subrings are ideals, Canad. J. Math. 20 (1968), 862871.
18 Najafizadeh, A., On the square submodule of a mixed module, Gen. Math. Notes 27(1) (2015), 18.
19 Rédei, L., Vollidealringe im weiteren Sinn. I, Acta Math. Acad. Sci. Hungar. 3 (1952), 243268.
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), 127130.
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Proceedings of the Edinburgh Mathematical Society
  • ISSN: 0013-0915
  • EISSN: 1464-3839
  • URL: /core/journals/proceedings-of-the-edinburgh-mathematical-society
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