Annihilator of Generalized Derivations with Power Values in Rings and Algebras

Year 2020, Volume: 8 Issue: 2, 65 - 70, 15.10.2020

Md Hamidur Rahaman

https://doi.org/10.36753/mathenot.631172

Abstract

Let $\mathcal{F}, \mathcal{G}$ be two  generalized derivations of prime ring $\mathcal{R}$ with characteristic different from 2 with associated derivations $d_1$ and $d_2$ respectively. We use the symbols  $\mathcal{C}=\mathcal{Z(U)}$ and  $\mathcal{U}$ to denote the  the extended centroid of $R$ and Utumi ring of quotient of $\mathcal{R}$ respectively. Let $0\neq a \in \mathcal{R}$ and $\mathcal{F}$ and $\mathcal{G}$ satisfy $a\{(\mathcal{F}(xy)+\mathcal{G}(yx))^m-[x,y]^n\}=0$ for all $x, y\in \mathcal{J}$, a nonzero ideal, where $m$ and $n$ are natural numbers. Then either $\mathcal{R}$ is commutative or there exists $c$, $b\in \mathcal{U}$ such that $\mathcal{F}$(x) = cx and $\mathcal{G}$(x) = bx for all x ∈ R. 

Keywords

Semiprime rings, Generalized derivations, extended centroid

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