A Work on Generalized Reverse Derivation and Skew-Derivation on Prime Near-Rings

This paper investigates the analysis of prime near-ring by explore the detailed structure of the generalized derivations that satisfy some specific assumptions. Let N be a prime near-rings and G be a generalized reverse derivative associated with mapping d on N. An additive mapping d:N→N is said to be a derivation on N if d(xy)=d(x)y+xd(y) for all x,y ∈N. A mapping G: N→N associated with derivation d is called a generalized derivation on N if G(xy)=G(x)y+xd(y) for all x,y∈N. Also, a mapping d: N→N is said to be a reverse derivation on N if d(xy)=d(y)x+yd(x) for all x,y ∈N and a mapping G:N→N associated with reverse derivation d is said to be a generalized reverse derivation on N if G(xy)=G(y)x+yd(x) for all x,y∈N. We prove some results on commutativity of prime near-rings involving generalized reverse derivations. In addition, we prove that; for prime near-rings N, if d(x)d(y)±xy=0 for all x,y∈N then d=0 where d is a skew- derivation associated with an automorphism β∶ N→N. Furthermore, for a prime near-ring N with generalized derivative G associated with mapping d on N, if G(x)G(y)±xy=0 for all x,y∈G then d=0.