Skip to main content

Some Classes of Hopfian Abelian Groups


This paper describes the Hopfian groups in some classes of Abelian groups. A complete description of Hopfian divisible groups is presented; on the basis of this description, the study of Hopficity of arbitrary Abelian groups is reduced to the study of Hopficity of reduced groups. Direct sums of cyclic groups that are Hopfian groups are characterized. Finally, the paper looks at specific examples of rings whose additive groups are Hopfian.

This is a preview of subscription content, access via your institution.


  1. 1.

    E. V. Kaygorodov, “About Hopficity of additive groups of some rings,” in: Modern Problems of Mathematics and Mechanics, Proc. Third National Youth Conf., Tomsk (2012), pp. 19–22.

  2. 2.

    E. V. Kaygorodov, “Hopfian Abelian groups,” Tomsk State Univ. J. Math. Mech., No. 2 (18), 5–12 (2012).

  3. 3.

    P. A. Krylov, A. V. Mikhalev, and A. A. Tuganbaev, Endomorphism Rings of Abelian Groups [in Russian], Faktorial, Moscow (2006).

    Google Scholar 

  4. 4.

    P. A. Krylov, A. A. Tuganbaev, and A. R. Chekhlov, Exercises on the Theory of Groups, Rings and Fields [in Russian], Tomsk. Gos. Univ., Tomsk (2008).

    Google Scholar 

Download references

Author information



Corresponding author

Correspondence to E. V. Kaygorodov.

Additional information

Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 17, No. 8, pp. 59–61, 2011/12.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Kaygorodov, E.V. Some Classes of Hopfian Abelian Groups. J Math Sci 197, 623–624 (2014).

Download citation


  • Abelian Group
  • Cyclic Group
  • Additive Group
  • Commutative Ring
  • Algebraic System