Siberian Mathematical Journal

, Volume 60, Issue 4, pp 727–733 | Cite as

On Fully Idempotent Homomorphisms of Abelian Groups

  • A. R. ChekhlovEmail author


We provide some examples of irregular fully idempotent homomorphisms and study the pairs of abelian groups A and B for which the homomorphism group Hom(A, B) is fully idempotent. We show that if B is a torsion group or a mixed split group and if at least one of the groups A or B is divisible then the full idempotence of the homomorphism group implies its regularity. If at least one of the groups A or B is a reduced torsion-free group and their homomorphism groups are nonzero then the group is not fully idempotent. The study of fully idempotent groups Hom(A, A) comes down to reduced mixed groups A with dense elementary torsion part.


regular homomorphism fully idempotent homomorphism homomorphism group mixed group self-small group 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abyzov A. N., “Fully idempotent homomorphisms,” Russian Math. (Iz. VUZ), vol. 55, no. 1, 1–6 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Abyzov A. N. and Tuganbaev A. A., “Homomorphisms close to regular and their applications,” J. Math. Sci. (N. Y.), vol. 183, no. 3, 275–298 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Kasch F., “Regularity in Hom,” Algebra-Berichte, vol. 75, 1–11 (1996).MathSciNetzbMATHGoogle Scholar
  4. 4.
    Nicholson W. K. and Zhou Y., “Semiregular morphisms,” Comm. Algebra, vol. 34, no. 1, 219–233 (2006).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Kasch F. and Mader A., “Regularity and substructures of Hom,” Comm. Algebra, vol. 34, no. 4, 1459–1478 (2006).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Kasch F., “Regularity and substructures of Hom,” Appl. Categ. Structures, vol. 16, no. 1–2, 159–166 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Mader A., “Regularity in endomorphism rings,” Comm. Algebra, vol. 37, no. 8, 2823–2844 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hakmi H., “Regularity and semipotency of Hom,” Korean J. Math., vol. 22, no. 1, 151–167 (2014).CrossRefGoogle Scholar
  9. 9.
    Kasch F. and Mader A., Regularity and Substructures of Hom, Birkhäuser Verlag, Basel, Boston, and Berlin (2009).zbMATHGoogle Scholar
  10. 10.
    Fuchs L., Infinite Abelian Groups. 2 vols, Academic Press, New York and London (1970, 1973).zbMATHGoogle Scholar
  11. 11.
    Rangaswamy K. M., “Abelian groups with endomorphism images of special types,” J. Algebra, vol. 6, no. 3, 271–280 (1967).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fuchs L. and Rangaswamy K. M., “On generalized regular rings,” Math. Z., Bd 107, 71–81 (1968).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Krylov P. A., “Mixed abelian groups as modules over their endomorphism rings,” Fund. Prikl. Mat., vol. 6, no. 3, 793–812 (2000).MathSciNetzbMATHGoogle Scholar
  14. 14.
    Krylov P. A. and Pakhomova E. G., “Abelian groups and regular modules,” Math. Notes, vol. 69, no. 3, 364–372 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Glaz S. and Wickless W., “Regular and principal projective endomorphism rings of mixed abelian groups,” Comm. Algebra, vol. 22, no. 4, 1161–1176 (1994).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Albrecht U. F., Goeters H. P., and Wickless W., “The flat dimension of mixed abelian groups as E-modules,” Rocky Mountain J. Math., vol. 25, no. 2, 569–590 (1995).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Fomin A. and Wickless W., “Quotient divisible abelian groups,” Proc. Amer. Math. Soc., vol. 126, no. 1, 45–52 (1998).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Fomin A. and Wickless W., “Self-small mixed abelian groups G with G/T(G) finite rank divisible,” Comm. Algebra, vol. 26, no. 11, 3563–3580 (1998).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Yakovlev A. V. and N’Famara Kamara, “Mixed abelian groups of finite rank and their direct decompositions,” Vestnik St. Petersburg Univ. Math., vol. 2, no. 26, 50–53 (1993).MathSciNetzbMATHGoogle Scholar
  20. 20.
    Yakovlev A. V., “Duality of the categories of torsion-free Abelian groups of finite rank and quotient divisible Abelian groups,” J. Math. Sci. (N. Y.), vol. 171, no. 3, 416–420 (2010).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Inc. 2019

Authors and Affiliations

  1. 1.Tomsk State UniversityTomskRussia

Personalised recommendations