Generalized skew-derivations as generalizations of Jordan homomorphisms in prime rings

Abstract

Let R be a noncommutative prime ring of characteristic different from 2, Q_r be the right Martindale quotient ring of R and $C=Z(Q_r)$ be the extended centroid of R. Suppose that $f(x_1,\dots ,x_n)$ is a noncentral multilinear polynomial over C and $0\ne F,$ G are two generalized skew-derivations of R associated to the same automorphism α. If $$F(u^2)=F(u)G(u)$$ for all $u\in f(R),$ then one of the following holds:

1. There exist $p\in Q_r$ and $\lambda \in C$ such that $F(x)=\lambda \alpha (x)$ and $G(x)=\alpha (x)$ for all $x\in R.$

2. There exists $a\in Q_r$ such that F(x) = ax for all $x\in R$ and G = Id, the identity map.

3. There exist $a,b\in Q_r$ such that F(x) = ax and G(x) = xb for all $x\in R,$ with $a(b-1)=0$ and $f{(x_1,\dots ,x_n)}^2$ is central valued on R.

Keywords:

• Derivation
• generalized derivation
• generalized skew derivation
• prime ring

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 16W25
• Secondary: 16N60
• 16R50

Acknowledgments

The authors are thankful to the anonymous referees for their very carefully reading and giving constructive comments that lead to significant improvement in the readability of the paper.