Characterization of $n$-Jordan Homomorphisms on Rings

Abstract

In this paper, we prove that if $\varphi:\mathcal{R}\longrightarrow\mathcal{R}'$ is an $n$-Jordan homomorphism, where $\mathcal{R}$ has a unit $e$, then the map $a\longmapsto \varphi(e)^{n-2}\varphi(a)$ is a Jordan homomorphism.  As a consequence we show, under special hypotheses, that each $n$-Jordan homomorphism is an $n$-homomorphism.