Some properties of prime near-rings with $(\sigma,\tau)$-derivation

Some results by Bell and Mason on commutativity in near-rings are generalized. Let $N$ be a prime right near-ring with multiplicative center $Z$ and let $D$ be a $(\sigma,\tau)$-derivation on $N$ such that $\sigma D=D\sigma$ and $\tau D=D\tau$. The following results are proved: (i) If $D(N)\subset Z$ or $[D(N),D(N)]=0$ or $[D(N),D(N)]_{\sigma,\tau}=0$ then $(N,+)$ is abelian; (ii) If $D(xy)=D(x)D(y)$ or $D(xy)=D(y)D(x)$ for all $x,y\in N$ then $D=0$.