A characterization of $b$-generalized skew derivations on lie ideal of a prime ring

Abstract

Let $R$ be a prime ring of characteristic different from $2$, $U$ be its Utumi quotient ring, $C$ be its extended centroid and $L$ be a non-central Lie ideal of $R$. Suppose $F$ and $G$ are two non-zero $b$-generalized skew derivations of $R$ associated with the same automorphism $\alpha$ such that $$puF(u)+F(u)uq=G(u^2),\text{with}p+q\notin C,\text{for all u∈L. }$$ This paper aims to study the complete structure of the $\mathrm{b}$-generalized skew derivations $F$ and $G$.