A Note on the Square Subgroups of Decomposable Torsion-Free Abelian Groups of Rank Three

Woronowicz, Mateusz

doi:10.1515/amsil-2017-0009

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.
Search in Google Scholar Back to article
[2] Aghdam A.M., Rings on indecomposable torsion free groups of rank two, Int. Math. Forum 1 (2006), no. 3, 141-146.
Search in Google Scholar Back to article
[3] Aghdam A.M., Najafizadeh A., Square subgroups of rank two Abelian groups, Colloq. Math. 117 (2009), no. 1, 19-28.
Search in Google Scholar Back to article
[4] Aghdam A.M., Najafizadeh A., Square submodule of a module, Mediterr. J. Math. 7 (2010), no. 2, 195-207.
Search in Google Scholar Back to article
[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.
Search in Google Scholar Back to article
[6] Andruszkiewicz R.R., Woronowicz M., Some new results for the square subgroup of an abelian group, Comm. Algebra 44 (2016), no. 6, 2351-2361.10.1080/00927872.2015.1044107
Open DOI Search in Google Scholar Back to article
[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.
Search in Google Scholar Back to article
[8] Andruszkiewicz R.R., Woronowicz M., On additive groups of associative and commutative rings, Quaest. Math. 40 (2017), no. 4, 527-537.
Search in Google Scholar Back to article
[9] Andruszkiewicz R.R., Woronowicz M., On the square subgroup of a mixed SI-group, Proc. Edinburgh Math. Soc. 61 (2018), no. 1, 295-304.
Search in Google Scholar Back to article
[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.
Search in Google Scholar Back to article
[11] Feigelstock S., On the type set of groups and nilpotence, Comment. Math. Univ. St. Pauli 25 (1976), 159-165.
Search in Google Scholar Back to article
[12] Feigelstock S., The absolute annihilator of an abelian group modulo a subgroup, Publ. Math. Debrecen 23 (1976), 221-224.
Search in Google Scholar Back to article
[13] Feigelstock S., Additive groups of rings, Vol. 1, Pitman Advanced Publishing Program, Boston, 1983.
Search in Google Scholar Back to article
[14] Fuchs L., Infinite abelian groups, Vol. 1, Academic Press, New York, London, 1970.
Search in Google Scholar Back to article
[15] Fuchs L., Infinite abelian groups, Vol. 2, Academic Press, New York, 1973.
Search in Google Scholar Back to article
[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.
Search in Google Scholar Back to article
[17] Najafizadeh A., On the square submodule of a mixed module, Gen. Math. Notes 27 (2015), no. 1, 1-8.
Search in Google Scholar Back to article
[18] Stratton A.E., Webb M.C., Abelian groups, nil modulo a subgroup, need not have nil quotient group, Publ. Math. Debrecen 27 (1980), 127-130.10.5486/PMD.1980.27.1-2.16
Search in Google Scholar Back to article
[19] Woronowicz M., A note on additive groups of some specific associative rings, Ann. Math. Sil. 30 (2016), 219-229.
Search in Google Scholar Back to article

A Note on the Square Subgroups of Decomposable Torsion-Free Abelian Groups of Rank Three

Abstract

Paradigm

My account