## Abstract

A hypothesis stated in [16] is confirmed for the case of associative rings. The answers to some questions posed in the mentioned paper are also given. The square subgroup of a completely decomposable torsion-free abelian group is described (in both cases of associative and general rings). It is shown that for any such a group *A*, the quotient group modulo the square subgroup of *A* is a nil-group. Some results listed in [16] are generalized and corrected. Moreover, it is proved that for a given abelian group *A*, the square subgroup of *A* considered in the class of associative rings, is a characteristic subgroup of *A*.

## References

[1] Aghdam A.M., Square subgroup of an Abelian group, Acta. Sci. Math. 51 (1987), 343–348.

[2] Aghdam A.M., Rings on indecomposable torsion free groups of rank two, Int. Math. Forum 1 (2006), no. 3, 141–146.

[3] Aghdam A.M., Najafizadeh A., Square subgroups of rank two Abelian groups, Colloq. Math. 117 (2009), no. 1, 19–28.

[4] Aghdam A.M., Najafizadeh A., Square submodule of a module, Mediterr. J. Math. 7 (2010), no. 2, 195–207.

[5] Aghdam A.M., Najafizadeh A., On the indecomposable torsion-free abelian groups of rank two, Rocky Mountain J. Math. 42 (2012), no. 2, 425–438.

6] Andruszkiewicz R.R., Woronowicz M., Some new results for the square subgroup of an abelian group, Comm. Algebra 44 (2016), no. 6, 2351–2361.

[7] Andruszkiewicz R.R., Woronowicz M., A torsion-free abelian group exists whose quotient group modulo the square subgroup is not a nil-group, Bull. Aust. Math. Soc. 94 (2016), no. 3, 449–456.

[8] Andruszkiewicz R.R., Woronowicz M., On additive groups of associative and commutative rings, Quaest. Math. (2017), DOI: 10.2989/16073606.2017.1302019.

[9] Andruszkiewicz R.R., Woronowicz M., On the square subgroup of a mixed SI-group, Proc. Edinburgh Math. Soc. (2017). To appear.

[10] Beaumont R.A., Wisner R.J., Rings with additive group which is a torsion-free group of rank two, Acta. Sci. Math. Szeged 20 (1959), 105–116.

[11] Feigelstock S., On the type set of groups and nilpotence, Comment. Math. Univ. St. Pauli 25 (1976), 159–165.

[12] Feigelstock S., The absolute annihilator of an abelian group modulo a subgroup, Publ. Math. Debrecen 23 (1976), 221–224.

[13] Feigelstock S., Additive groups of rings, Vol. 1, Pitman Advanced Publishing Program, Boston, 1983.

[14] Fuchs L., Infinite abelian groups, Vol. 1, Academic Press, New York, London, 1970.

[15] Fuchs L., Infinite abelian groups, Vol. 2, Academic Press, New York, 1973.

[16] Hasani F., Karimi F., Najafizadeh A., Sadeghi M.Y., On the square subgroups of decomposable torsion-free abelian groups of rank three, Adv. Pure Appl. Math. 7 (2016), no. 4, 259–265.

[17] Najafizadeh A., On the square submodule of a mixed module, Gen. Math. Notes 27 (2015), no. 1, 1–8.

[18] Stratton A.E., Webb M.C., Abelian groups, nil modulo a subgroup, need not have nil quotient group, Publ. Math. Debrecen 27 (1980), 127–130.

[19] Woronowicz M., A note on additive groups of some specific associative rings, Ann. Math. Sil. 30 (2016), 219–229.