Commutativity of $\Gamma$-Generalized Boolean Semirings with Derivations

On Generalized Derivations and Commutativity of Prime Rings with Involution

Let R be a ring with involution ′∗′. A map δ of the ring R into itself is called a derivation if δ(xy) = δ(x)y + xδ(y) for all x, y ∈ R. An additive map F : R → R is called a generalized derivation on R if F(xy) = F(x)y + xδ(y) for all x, y ∈ R, Permanent address: Department of Mathematics, Faculty of Science, Aligarh Muslim University, Aligarh202002, India 292 Shakir Ali and Husain Alhazmi whe...

Generalized Derivations in Semiprime Gamma Rings

Generalized Derivations on Semiprime Gamma Rings with Involution

An extensive generalized concept of classical ring set forth the notion of a gamma ring theory. As an emerging field of research, the research work of classical ring theory to the gamma ring theory has been drawn interest of many algebraists and prominent mathematicians over the world to determine many basic properties of gamma ring and to enrich the world of algebra. The different researchers ...

On Prime-Gamma-Near-Rings with Generalized Derivations

Copyright q 2012 Kalyan Kumar Dey et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Let N be a 2-torsion free prime Γ-near-ring with center ZN. Let f, d and g, h be two generalized derivations on N. We prove the following res...

On derivations and commutativity in prime rings

Let R be a prime ring of characteristic different from 2, d a nonzero derivation of R, and I a nonzero right ideal of R such that [[d(x), x], [d(y), y]] = 0, for all x, y ∈ I. We prove that if [I, I]I ≠ 0, then d(I)I = 0. 1. Introduction. Let R be a prime ring and d a nonzero derivation of R. Define [x, y] 1 = [x, y] = xy − yx, then an Engel condition is a polynomial [x, y] k = [[x, y] k−1 ,y]