Characterization of $n$-Jordan Homomorphisms on Rings

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.

Characterization of mixed n-Jordan and pseudo n-Jordan homomorphisms

In this article, a new notion of n-Jordan homomorphism namely the mixed n-Jordan homomorphism is introduced. It is proved that how a mixed (n + 1)-Jordan homomorphism can be a mixed n-Jordan homomorphism and vice versa. By means of some examples, it is shown that the mixed n-Jordan homomorphisms are different from the n-Jordan homomorphisms and the pseudo n-Jordan homomorphisms. As a consequence, it shown that every mixed Jordan homomorphism from Banach algebra A into commutative semisimple Banach algebra B is automatically continuous. Under some mild conditions, every unital pseudo 3-Jordan homomorphism is a homomorphism.

U- S Jordan Homomorphisim of Inverse Semirings

Let S be an inverse semiring, and U be an ideal of S. In this paper, we introduce the concept of U-S Jordan homomorphism of inverse semirings, and extend the result of Herstein on Jordan homomorphisms in inverse semirings.