Idempotent values of commutators involving generalized derivations

Gurninder S. Sandhu ; Shakir Ali

Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 15 (2022) no. 3, pp. 356-365

Résumé

In the present article, we characterize generalized derivations and left multipliers of prime rings involving commutators with idempotent values. Precisely, we prove that if a prime ring of characteristic different from $2$ admits a generalized derivation $G$ with an associative nonzero derivation $g$ of $R$ such that $[G(u),u]^{n}=[G(u),u]$ for all $u\in\{[x,y]:x,y\in L\},$ where $L$ a noncentral Lie ideal of $R$ and $n>1$ is a fixed integer, then one of the following holds: $R$ satisfies $s_{4}$ and there exists $\lambda\in C,$ the extended centroid of $R$ such that $G(x)=ax+xa+\lambda x$ for all $x\in R,$ where $a\in U,$ the Utumi quotient ring of $R,$ there exists $\gamma\in C$ such that $G(x)=\gamma x$ for all $x\in R.$ As an application, we describe the structure of left multipliers of prime rings satisfying the condition $([T^m (u),u] )^{n}=[T^m (u),u]$ for all $u\in \{[x,y]: x,y\in L\},$ where $m,n>1$ are fixed integers. In the end, we give an example showing that the hypothesis of our main theorem is not redundant.

Détail

Keywords: prime ring, Lie ideal, generalized derivation, GPI.