Annihilator of Generalized Derivations with Power Values in Rings and Algebras

Authors : Md Hamidur RAHAMAN

Pages : 65-70

Doi:10.36753/mathenot.631172

View : 27 | Download : 7

Publication Date : 2020-10-15

Article Type : Research Paper

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  $\mathcal{C}=\mathcal{Zinsert ignore into journalissuearticles values(U);}$ 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 $a\{insert ignore into journalissuearticles values(\mathcal{F}insert ignore into journalissuearticles values(xy);+\mathcal{G}insert ignore into journalissuearticles values(yx););^m-[x,y]^n\}=0$ 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  $\mathcal{F}$ insert ignore into journalissuearticles values(x); = cx and  $\mathcal{G}$ insert ignore into journalissuearticles values(x); = bx for all x ∈ R. 
Keywords : Semiprime rings, Generalized derivations, extended centroid

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