Derivations in prime near-rings

Abstract:

Let N be a prime near-ring with center Z. The purpose of this paper is to study derivations on N. We show two main results: (1) Let N be 2-torsion-free, and let ${D_1}$ and ${D_2}$ be derivations on N such that ${D_1}{D_2}$ is also a derivation. Then either ${D_1}$ or ${D_2}$ is zero if and only if $[{D_1}(x),{D_2}(y)] = 0$ for all $x,y \in N$. (2) Let n be an integer $\geq 1$, N be n!-torsion-free, and D a derivation with ${D^n}(N) = \{ 0\}$. Then $D(Z) = \{ 0\}$.