Relations Between n-Jordan Homomorphisms and n-Homomorphisms


For \(n\ge 2\), an additive map f between two rings A and B is called an n-Jordan homomorphism, or an n-homomorphism if \(f(a^n)=f(a)^n\), for all \(a\in A\), or \(f(a_1a_2\cdots a_n)=f(a_1)f(a_2)\cdots f(a_n)\), for all \(a_1,a_2,\ldots ,a_n\in A\), respectively. In particular, if \(n=2\) then f is simply called a Jordan homomorphism or a homomorphism, respectively. The notion of n-Jordan homomorphism between rings was introduced in 1956 by Herstein and the concept of n-homomorphism between algebras was introduced in 2005 by Hejazian et al. Properties of n-Jordan homomorphisms as well as n-homomorphisms have been studied by many authors since then. One of the main questions is that, “under what conditions n- Jordan homomorphisms are n-homomorphism?”. Another natural question is that “under what conditions certain properties of homomorphisms may be extended to n-homomorphisms”. We provide conditions under which these questions have affirmative answers. We also study the continuity problem for n-Jordan homomorphisms on Banach algebras, while extending some known results in this field.

This is a preview of subscription content, log in to check access.


  1. 1.

    Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions with formulas. Graphs and Mathematical Tables, US Department of Commerce, Washington, DC (1964)

  2. 2.

    An, G.: Characterization of \(n\)-Jordan homomorphisms. Linear Multilinear Algebra 66(4), 1–10 (2017)

    MathSciNet  Google Scholar 

  3. 3.

    Bodaghi, A., Shojaee, B.: \(n\)-Jordan homomorphisms on \(C^{*}\)-algebras. JLTA 01(01), 1–7 (2012)

    MATH  Google Scholar 

  4. 4.

    Bodaghi, A., Inceboz H.: \(n\)-Jordan homomorphisms on commutative algebras. Acta Math. Uni. Comenianae, Vol. LXZZVII, 1, 141–146 (2018)

  5. 5.

    Bračič, J., Moslehian, M.S.: On automatic continuity of 3-homomorphisms on Banach algebras. Bull. Malays. Math. Sci. Soc. (2) 30(2), 195–200 (2007)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Eshaghi, Gordji M.: \(n\)-Jordan homomorphisms. Bull. Aust. Math. Soc. 80, 159–164 (2009)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Eshaghi Gordji, M., Jabbari, A., Karapinar, E.: Automatic continuity of surjective \(n\)-homomorphisms on Banach algebras. Bull. Iran Math. Soc. 41(5), 1207–1211 (2015)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Gselmann, E.: On approximate \(n\)-Jordan homomorphisms. Ann. Math. Sil. 28, 47–58 (2014)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Hejazian, S., Mirzavaziri, M., Moslehian, M.S.: \(n\)-homomorphisms. Bull. Iran. Math. Soc. 31(1), 13–23 (2009)

    MATH  Google Scholar 

  10. 10.

    Herstein, I.N.: Jordan homomorphisms. Trans. Am. Math. Soc. 81, 331–341 (1956)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Honary, T.G., Shayanpour, H.: Automatic continuity of n-homomorphisms between Banach algebras. Quaest. Math. 33, 189–196 (2010)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Jacobson, N., Rickart, C.E.: Jordan homomorphisms of rings. Trans. Am. Math. Soc. 69, 479–502 (1950)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Kaplansky, I.: Semi-automorphisms of rings. Duke Math. J. 14, 521–525 (1947)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Lee Y. H.: Stability of \(n\)-Jordan homomorphisms from a normed algebra to a Banach algebra. Abstr. Appl. Anal. ID 691025, p. 5 (2013)

  15. 15.

    Palmer, T.: Banach algebras and the general theory of *-algebras, vol. I. Cambridge University Press, Cambridge (1994)

    Google Scholar 

  16. 16.

    Żelazko, W.: A characterization of multiplicative linear functionals in complex Banach algebras. Stud. Math. 30, 83–85 (1968)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Zivari-Kazempour, A.: A characterization of \(3\)-Jordan homomorphisms on Banach algebras. Bull. Aust. Math. Soc. 93(2), 301–306 (2016)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Zivari-Kazempour, A.: A characterization of Jordan and \(5\)-Jordan homomorphisms between Banach algebras. Asian Eur. J. Math. 11(2), 1–10 (2018)

    MathSciNet  Article  Google Scholar 

Download references


The authors would like to thank the referee whose comments helped us to improve this article.

Author information



Corresponding author

Correspondence to Taher Ghasemi Honary.

Ethics declarations

Conflict of Interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by Mohammad B. Asadi.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Honary, T.G., Hosseinzadeh, H. & Mohammadi, S. Relations Between n-Jordan Homomorphisms and n-Homomorphisms. Bull. Iran. Math. Soc. (2020).

Download citation


  • Homomorphism
  • Jordan homomorphism
  • n-Jordan homomorphism
  • n-Homomorphism
  • Banach algebra
  • Automatic continuity

Mathematics Subject Classification

  • 13B10
  • 47C05
  • 47B48
  • 46H40