$(\theta,\phi)$-derivations as homomorphisms or as anti-homomorphisms on a near ring

Abstract

Let $N$ be a near ring. An additive mapping $d:N\longrightarrow N$ is said to be a $(\theta,\phi)$-derivation on $N$ if there exist mappings $\theta,\phi:N\longrightarrow N$ such that$d(xy)=\theta(x)d(y)+d(x)\phi(y)$ holds for all $x,y \in N$. In the context of 3-prime and 3-semiprime nearrings, we show that for suitably-restricted $\theta$ and $\phi$, there exist no nonzero $(\theta,\phi)$-derivations which act as a homomorphism or an anti-homomorphism on $N$ or a nonzero semigroup ideal of $N$.