Jordan Product Preserving Generalized Skew Derivations on Lie Ideals

Abstract

Let R be a non-commutative prime ring of characteristic different from 2, $Q_r$ be its right Martindale quotient ring and C be its extended centroid, L a non-central Lie ideal of R and F and G non-zero generalized skew derivations of R such that $F(x)G(y)+F(y)G(x)=x\circ y$, for all $x,y\in L$. Then one of the following holds:

• 1. there exists $a,b\in Q_r$ such that $F(x)=xa$ and $G(x)=bx$, for any $x\in R$, with $ab=1_{C}$;

• 2. R satisfies $s_4(x_1,\ldots ,x_4)$ the standard polynomial identity on 4 non-commuting variables and there exist $a,b\in Q_r$ such that $F(x)=ax$ and $G(x)=xb$, for any $x\in R$, with $ab=1_{C}$;

• 3. R satisfies $s_4(x_1,\ldots ,x_4)$ and there exist an invertible element $q\in Q_r$ and $0\ne \beta \in C$ such that $F(x)=\beta qxq^{-1}$ and $G(x)=\beta ^{-1}qxq^{-1}$, for any $x\in R$.