A Note on Additive Groups of Some Specific Associative Rings

Open access

Abstract

Almost complete description of abelian groups (A, +, 0) such that every associative ring R with the additive group A satisfies the condition: every subgroup of A is an ideal of R, is given. Some new results for SR-groups in the case of associative rings are also achieved. The characterization of abelian torsion-free groups of rank one and their direct sums which are not nil-groups is complemented using only elementary methods.

References

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

  • [2] Aghdam A.M., Karimi F., Najafizadeh A., On the subgroups of torsion-free groups which are subrings in every ring, Ital. J. Pure Appl. Math. 31 (2013), 63–76.

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

  • [4] Andruszkiewicz R.R., Woronowicz M., On associative ring multiplication on abelian mixed groups, Comm. Algebra 42 (2014), no. 9, 3760–3767.

  • [5] Andruszkiewicz R.R., Woronowicz M., On SI-groups, Bull. Aust. Math. Soc. 91 (2015), 92–103.

  • [6] Chekhlov A.R., On abelian groups, in which all subgroups are ideals, Vestn. Tomsk. Gos. Univ. Mat. Mekh. (2009), no. 3(7), 64–67

  • [7] Feigelstock S., Additive groups of rings. Vol. I, Pitman Advanced Publishing Program, Boston, 1983.

  • [8] Feigelstock S., Additive groups of rings whose subrings are ideals, Bull. Austral. Math. Soc. 55 (1997), 477–481.

  • [9] Feigelstock S., Rings in which a power of each element is an integral multiple of the element, Archiv der Math. 32 (1979), 101–103.

  • [10] Fuchs L., Infinite abelian groups. Vol. I, Academic Press, New York-London, 1970.

  • [11] Fuchs L., Infinite abelian groups. Vol. II, Academic Press, New York-London, 1973.

  • [12] Kompantseva E.I., Absolute nil-ideals of Abelian groups, Fundam. Prikl. Mat. 17 (2012), no. 8, 63–76.

  • [13] Kompantseva E.I., Abelian dqt-groups and rings on them, Fundam. Prikl. Mat. 18 (2013), no. 3, 53–67.

  • [14] O’Neill J.D., Rings whose additive subgroup are subrings, Pacific J. Math. 66 (1976), no. 2, 509–522.

  • [15] Pham Thi Thu Thuy, Torsion abelian RAI-groups, J. Math. Sci. (N. Y.) 197 (2014), no. 5, 658–678.

  • [16] Pham Thi Thu Thuy, Torsion abelian afi-groups, J. Math. Sci. (N. Y.) 197 (2014), no. 5, 679–683.

Journal Information


Mathematical Citation Quotient (MCQ) 2016: 0.10

Target Group

researchers in all branches of pure and applied mathematics

Metrics

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 28 28 25
PDF Downloads 4 4 3