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

TL;DR

This paper characterizes the structure of b-generalized skew derivations on Lie ideals within prime rings, revealing their algebraic properties and relationships under specific conditions.

Contribution

It provides a complete structural description of b-generalized skew derivations on Lie ideals in prime rings, a topic not previously fully explored.

Findings

Derived explicit forms of b-generalized skew derivations

Established conditions under which these derivations are related

Extended understanding of derivations in prime ring theory

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 $$puF(u) + F(u)uq = G(u^2),\ \text{with} \ p + q \notin C,~~ \text{for all $u \in L$. }$$ This paper aims to study the complete structure of the $\mathrm{b}$-generalized skew derivations $F$ and $G$.