A note on n-Jordan homomorphisms

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.}