Hypercommuting conditions of b-generalized skew derivations on Lie ideals in prime rings

期刊论文
发表日期：

2024年9月16日

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B. Dhara,
G. S. Sandhu

摘要

Let R be any non-commutative prime ring of char $(R)\ne 2$ $(R) \neq 2$ , L a non-central Lie ideal of R and F, G be b-generalized skew derivations of R. Suppose that $[F(u)u-uG(u), u]_n=0$ ${[F (u) u - u G (u), u]}_{n} = 0$ for all $u\in L$ $u \in L$ and for some fixed integer $n\ge 1$ $n \geq 1$ , then one of the following assertions holds: (1)

there exist $a'',b''\in Q_r$ $a^{''}, b^{''} \in Q_{r}$ such that $F(x)=xa''$ $F (x) = x a^{''}$ , $G(x)=b''x$ $G (x) = b^{''} x$ for all $x\in R$ $x \in R$ with $a''-b''\in C$ $a^{''} - b^{''} \in C$ ;

(2)

$R\subseteq M_2(K),$ $R \subseteq M_{2} (K),$ the algebra of $2\times 2$ $2 \times 2$ matrices over a field K and

either K is a finite field;

or there exists $\lambda \in C$ $λ \in C$ such that $(F+G)(x)=\lambda x$ $(F + G) (x) = λ x$ for all $x\in R$ $x \in R$ ;

or there exists $\lambda \in C$ $λ \in C$ and $h\in Q_{r}$ $h \in Q_{r}$ such that $(F+G)(x)=hx+xh+\lambda x$ $(F + G) (x) = h x + x h + λ x$ for all $x\in R$ $x \in R$ .

The above result, naturally improves the recent result obtained by Carini et al. in [4].