A Note on the Commutativity of Prime Near-rings

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