Commutators and generalized derivations acting on Lie ideals in prime rings

期刊论文
发表日期：

2024年5月23日

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Basudeb Dhara

摘要

Let R be a prime ring of char $(R)\ne 2, 3$ $(R) \neq 2, 3$ and L a noncentral Lie ideal of R. Let U be the Utumi quotient ring of R and $C=Z(U)$ $C = Z (U)$ be the extended centroid of R. Suppose that F, G, H are three generalized derivations of R such that $[F(u),u]G(u)+u[H(u),u]=0$ $[F (u), u] G (u) + u [H (u), u] = 0$ for all $u\in L$ $u \in L$ . Then either R satisfies standard polynomial $s_4(x_1,x_2,x_3,x_4)$ $s_{4} (x_{1}, x_{2}, x_{3}, x_{4})$ or one of the following holds: 1.

There exist $\alpha , \beta \in C$ $α, β \in C$ such that $F(x)= \alpha x$ $F (x) = α x$ and $H(x)= \beta x$ $H (x) = β x$ for all $ x\in R$ $x \in R$ ;

There exists $\beta \in C$ $β \in C$ such that $G(x)=0$ $G (x) = 0$ , $H(x)=\beta x$ $H (x) = β x$ for all $ x\in R$ $x \in R$ ;

There exist $a,b\in U$ $a, b \in U$ and $0\ne \mu \in C$ $0 \neq μ \in C$ such that $F(x)=xa$ $F (x) = x a$ , $G(x)=\mu x$ $G (x) = μ x$ , $H(x)=bx$ $H (x) = b x$ for all $ x\in R$ $x \in R$ with $\mu a+b\in C$ $μ a + b \in C$ .