A Note on the Commutativity of Prime Near-rings

Abstract

Let N be a prime near-ring. We show two main results on the commutativity of N: (1) If there exist k, l ∈ ℕ such that N admits a generalized derivation D satisfying either D([x,y])=xk[x,y]xlfor all x, y ∈ N or D([x,y])=-xk[x,y]xlfor all x, y ∈ N, then N is a commutative ring. (2) If there exist k, l ∈ ℕ such that N admits a generalized derivation D satisfying either D(x ◦ y)= xk(x ◦ y) xlfor all x, y ∈ N or D(x ◦ y)= -xk(x ◦ y) xlfor all x, y ∈ N, then N is a commutative ring. Moreover, some interesting relations between the prime graph and zero-divisor graph of N are studied.