A note on n-Jordan homomorphisms

  • M. El Azhari Ecole Normale Sup´erieure Avenue Oued Akreuch, Takaddoum, BP 5118, Rabat, Morocco
Keywords: Jordan homomorphism, homomorphism, n-Jordan homomorphism, n-homomorphism

Abstract

{Let $ A, B $ be two rings and $ n\geqslant 2 $ be an integer. An additive map $ h\colon A\rightarrow B $ is called an $n$-Jordan homomorphism if $ h(x^{n})=h(x)^{n} $ for all $ x\in A;$ $h$ is called an n-homomorphism or an anti-$n$-homomorphism if $ h(\prod_{i=1}^{n}x_{i})=\prod_{i=1}^{n} h(x_{i})$ or $ h(\prod_{i=1}^{n}x_{i})=\prod_{i=0}^{n-1} h(x_{n-i}),
$ respectively, for all $ x_{1},...,x_{n}\in A. $}

{We give the following variation of a theorem on n-Jordan homomorphisms due to I.N. Herstein: Let $n\geq 2$ be an integer and $h$ be an $n-$Jordan
homomorphism from a ring $A$ into a ring $B$ of characteristic greater than $n$.
Suppose further that $A$ has a unit $e$, then $h = h(e)\tau$, where $h(e)$ is in the centralizer of $h(A)$ and $\tau$ is a Jordan homomorphism.}

{By using this variation, we deduce the following result of G. An: Let $A$ and $B$
be two rings, where $A$ has a unit and $B$ is of characteristic greater than an integer $n \geq 2$. If every Jordan homomorphism from $A$ into $B$ is a homomorphism (anti-homomorphism), then every $n-$Jordan homomorphism from $A$ into $B$ is an $n$-homomorphism (anti-$n$-homomorphism).
As a consequence of an appropriate lemma, we also obtain the following result
of E. Gselmann: Let $A, B$ be two commutative rings and $B$ is of characteristic greater than an integer $n\geq 2$. Then every $n$-Jordan homomorphism from $A$ into
$B$ is an $n-$homomorphism.}

Author Biography

M. El Azhari, Ecole Normale Sup´erieure Avenue Oued Akreuch, Takaddoum, BP 5118, Rabat, Morocco

Ecole Normale Sup´erieure
Avenue Oued Akreuch, Takaddoum, BP 5118,
Rabat, Morocco

References

G. An, Characterizations of n-Jordan homomorphisms, Linear and Multilinear Algebra, 66 (2018), №4, 671–680. https://doi.org/10.1080/03081087.2017.131881

A. Bodaghi, H. Inceboz, n-Jordan homomorphisms on commutative algebras, Acta Math. Univ. Comenianae (N.S.), 87 (2018), №1, 141–146. http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/635/569

E. Gselmann, On approximate n-Jordan homomorphisms, Ann. Math. Sil., 28 (2014), 47–58. https://www.journals.us.edu.pl/index.php/AMSIL/article/view/13990

I.N. Herstein, Jordan homomorphisms, Trans. Amer. Math. Soc., 81 (1956), №2, 331–341. https://doi.org/10.2307/1992920

Y.-H. Lee, Stability of n-Jordan homomorphisms from a normed algebra to a Banach algebra, Abstr. Appl. Anal., 2013 (2013), Art. ID 691025, 5 p. https://doi.org/10.1155/2013/691025

Published
2024-09-15
How to Cite
El Azhari, M. (2024). A note on n-Jordan homomorphisms. Matematychni Studii, 62(1), 77-80. https://doi.org/10.30970/ms.62.1.77-80
Section
Articles