Annihilator of generalized derivations with power values in rings and Algebras

Md Hamidur RAHAMAN
2020 Mathematical sciences and applications e-notes  
Let F, G be two generalized derivations of prime ring R with characteristic different from 2 with associated derivations D 1 and D 2 respectively. We use the symbols C = Z(U) and U to denote the the extended centroid of R and Utumi ring of quotient of R respectively. Let 0 = a ∈ R and F and G satisfy a{(F(xy) + G(yx)) m − [x, y] n } = 0 for all x, y ∈ J , a nonzero ideal, where m and n are natural numbers. Then either R is commutative or there exists c, b ∈ U such that F(x) = cx and G(x) = bx
more » ... cx and G(x) = bx for all x ∈ R.
doi:10.36753/mathenot.631172 fatcat:lpp2mb2swfdxlfwg3gnvipxxxq