MMN-2450
On Lie ideals and symmetric generalized (α, β)-biderivation in prime ring
Nadeem ur Rehman; Shuliang Huang;Abstract
Let $\mathfrak{R}$ be a prime ring with char$\mathfrak{R} \neq 2$. A biadditive symmetric map $\Delta : \mathfrak{R} \times \mathfrak{R} \to \mathfrak{R}$ is
called symmetric $(\alpha, \beta)$-biderivation if, for any fixed $y \in \mathfrak{R}$, the map $x \mapsto \Delta(x, y)$ is a $(\alpha, \beta)$-derivation. A symmetric
biadditive map $\Gamma : \mathfrak{R} \times \mathfrak{R} \to \mathfrak{R}$ is a symmetric generalized $(\alpha, \beta)$-biderivation if for any fixed $y \in \mathfrak{R}$, the
map $x\mapsto \Gamma(x, y)$ is a generalized $(\alpha, \beta)$-derivation of $R$ associated with the $(\alpha, \beta)$-derivation $\Delta(., y)$. In the present
paper, we investigate the commutativity of a ring having a generalized $(\alpha, \beta)$-biderivation satisfying certain
algebraic conditions.
Vol. 20 (2019), No. 2, pp. 1175-1183